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Torus actions and their applications in topology and combinatorics
Victor M. Buchstab er Taras E. Panov

Author address:
Department of Mathematics and Mechanics, Moscow State University, 119899 Moscow RUSSIA

E-mail address :

buchstab@mech.math.msu.su

Department of Mathematics and Mechanics, Moscow State University, 119899 Moscow RUSSIA

E-mail address :

tpanov@mech.math.msu.su


1991 Mathematics Subject Classi cation. 52B70, 57Q15, 57R19, 14M25, 52B05, 13F55, 52C35. as a bridge connecting combinatorial and convex geometry with commutative and homological algebra, algebraic geometry, and topology. This link helps in understanding the geometry and topology of a space with torus action by studying the combinatorics of the space of orbits. Conversely, the most subtle properties of a combinatorial ob ject can be recovered by realizing it as the orbit structure for a proper manifold or complex acted on by a torus. The latter can be a symplectic manifold with Hamiltonian torus action, a toric variety or manifold, a subspace arrangement complement, etc., while the combinatorial ob jects include simplicial and cubical complexes, polytopes, and arrangements. This approach also provides a natural topological interpretation in terms of torus actions of many constructions from commutative and homological algebra used in combinatorics. The exposition centers around the theory of moment-angle complexes, providing an e ective way to study triangulations by methods of equivariant topology. The book includes many new and well-known open problems and would be suitable as a textbook. We hope that it will be useful for specialists both in topology and in combinatorics and will help to establish even tighter connections between the sub jects involved.
Abstract. Here, the study of torus actions on topological spaces is presented


Contents
Introduction Chapter 1.1. 1.2. 1.3. 1.4. 1.5. Chapter 2.1. 2.2. 2.3. 2.4. 2.5. Chapter 3.1. 3.2. 3.3. 3.4. 3.5. 1. Polytopes De nitions and main constructions Face vectors and Dehn{Sommerville equations The g-theorem Upper Bound and Lower Bound theorems Stanley{Reisner face rings of simple polytopes 2. Topology and combinatorics of simplicial complexes Abstract simplicial complexes and polyhedrons Basic P L topology, and operations with simplicial complexes Simplicial spheres Triangulated manifolds Bistellar moves 3. Commutative and homological algebra of simplicial complexes Stanley{Reisner face rings of simplicial complexes Cohen{Macaulay rings and complexes Homological algebra background Homological properties of face rings: Tor-algebras and Betti numbers Gorenstein complexes and Dehn{Sommerville equations 1 7 7 11 15 18 20 21 21 23 28 29 31 35 35 38 40 42 46 49 49 50 57 57 63 69 74 82 85 85 87 89

Chapter 4. Cubical complexes 4.1. De nitions and cubical maps 4.2. Cubical subdivisions of simple polytopes and simplicial complexes Chapter 5.1. 5.2. 5.3. 5. Toric and quasitoric manifolds Toric varieties Quasitoric manifolds Stably complex structures, and quasitoric representatives in cobordism classes 5.4. Combinatorial formulae for Hirzebruch genera of quasitoric manifolds 5.5. Classi cation problems 6. Moment-angle complexes Moment-angle manifolds ZP de ned by simple polytopes General moment-angle complexes ZK Cell decompositions of moment-angle complexes
vii

Chapter 6.1. 6.2. 6.3.


viii

CONTENTS

6.4. Moment-angle complexes corresponding to joins, connected sums and bistellar moves 92 6.5. Borel constructions and Davis{Januszkiewicz space 94 6.6. Walk around the construction of ZK : generalizations, analogues and additional comments 97 Chapter 7. Cohomology of moment-angle complexes and combinatorics of triangulated manifolds 101 7.1. The Eilenberg{Moore spectral sequence 101 7.2. Cohomology algebra of ZK 102 7.3. Bigraded Betti numbers of ZK : the case of general K 106 7.4. Bigraded Betti numbers of ZK : the case of spherical K 110 7.5. Partial quotients of ZP 113 117 7.6. Bigraded Poincare duality and Dehn{Sommerville equations Chapter 8. Cohomology rings of subspace arrangement complements 125 8.1. General arrangements and their complements 125 8.2. Coordinate subspace arrangements and the cohomology of ZK . 127 8.3. Diagonal subspace arrangements and the cohomology of ZK . 133 Bibliography 135 Index 141


Introduction
Torus actions on topological spaces is classical and one of the most developed elds in equivariant topology. Speci c problems connected with torus actions arise in different areas of mathematics and mathematical physics, which results in permanent interest in the theory, constant source of new applications and penetration of new ideas in topology. Many volumes devoted to particular aspects of this wide eld of mathematical knowledge are available. The topological approach is the sub ject of monograph 26] by G. Bredon. Monograph 9] by M. Audin deals with torus actions from the symplectic geometry viewpoint. The algebro-geometrical part of the study, known as the geometry of toric varieties or simply \toric geometry", is presented in several texts. These include V. Danilov's original survey article 46] and more recent monographs by T. Oda 105], W. Fulton 64] and G. Ewald 61]. The orbit space of a torus action carries a rich combinatorial structure. In many cases studying the combinatorics of the quotient is the easiest and the most e cient way to understand the topology of a toric space. This approach works in the opposite direction as well: the equivariant topology of a torus action sometimes helps to interpret and prove the most subtle combinatorial results topologically. In the most symmetric and regular cases (such as pro jective toric varieties or Hamiltonian torus actions on symplectic manifolds) the quotient can be identi ed with a convex polytope. More general toric spaces give rise to other combinatorial structures related with their quotients. Examples here include simplicial spheres, triangulated manifolds, general simplicial complexes, cubical complexes, subspace arrangements, etc. Combined applications of combinatorial, topological and algebro-geometrical methods stimulated intense development of toric geometry during the last three decades. This remarkable con uence of ideas enriched all the sub jects involved with a number of spectacular results. Another source of applications of topological and algebraic methods in combinatorics was provided by the theory of Stanley{ Reisner face rings and Cohen{Macaulay complexes, described in R. Stanley's monograph 128]. Our motivation was to broaden the existing bridge between torus actions and combinatorics by giving some new constructions of toric spaces, which naturally arise from combinatorial considerations. We also interpret many existing results in such a way that their relationships with combinatorics become more transparent. Traditionally, simplicial complexes, or triangulations, were used in topology as a tool for combinatorial treatment of topological invariants of spaces or manifolds. On the other hand, triangulations themselves can be regarded as particular structures, so the space of triangulations becomes the ob ject of study. The idea of considering the space of triangulations of a given manifold has been
1


2

INTRODUCTION

also motivated by some physical problems. One gets an e ective way of treatment of combinatorial results and problems concerning the number of faces in a triangulation by interpreting them as extremal value problems on the space of triangulations. We implement some of these ideas in our book as well, by constructing and investigating invariants of triangulations using the equivariant topology of toric spaces. The book is intended to be a systematic but elementary overview for the aspects of torus actions mostly related to combinatorics. However, our level of exposition is not balanced between topology and combinatorics. We do not assume any particular reader's knowledge in combinatorics, but in topology a basic knowledge of characteristic classes and spectral sequences techniques may be very helpful in the last chapters. All necessary information is contained, for instance, in S. Novikov's book 104]. We would recommend this book since it is reasonably concise, has a rather broad scope and pays much attention to the combinatorial aspects of topology. Nevertheless, we tried to provide necessary background material in the algebraic topology and hope that our book will be of interest to combinatorialists as well. A signi cant part of the text is devoted to the theory of moment-angle complexes, currently being developed by the authors. This study was inspired by paper 48] of M. Davis and T. Januszkiewicz, where a topological analogue of toric varieties was introduced. In their work, Davis and Januszkiewicz used a certain universal T m-space ZK , assigned to every simplicial complex on the vertex set m] = f1; : : : ; mg. In its turn, the de nition of ZK was motivated (see 47, x13]) by the construction of the Coxeter complex of a Coxeter group and its generalizations by E. Vinberg 137]. Our approach brings the space ZK to the center of attention. To each subset m] there is assigned a canonical T m-equivariant embedding (D2 )k T m?k 2 )m , where (D2 )m is the standard poly-disc in C m and k is the cardinality of . (D This correspondence extends to any simplicial complex K on m] and produces a canonical bi graded cell decomposition of the Davis{Januszkiewicz T m-space ZK , which we refer to as the moment-angle complex . There is also a more general version of moment-angle complexes, de ned for any cubical subcomplex in a unit cube (see section 4.2). The construction of ZK gives rise to a functor (see Proposition 7.12) from the category of simplicial complexes and inclusions to the category of T mspaces and equivariant maps. This functor induces a homomorphism between the standard simplicial chain complex of a simplicial pair (K1 ; K2) and the bigraded cellular chain complex of (ZK1 ; ZK2 ). The remarkable property of the functor is that it takes a simplicial Lefschetz pair (K1 ; K2 ) (i.e. a pair such that K1 n K2 is an open manifold) to another Lefschetz pair (of moment-angle complexes) in such a way that the fundamental cycle is mapped to the fundamental cycle. For instance, if K is a triangulated manifold, then the simplicial pair (K; ?) is mapped to the pair (ZK ; Z? ), where Z? = T m and ZK n Z? is an open manifold. Studying the functor K 7! ZK , one interprets the combinatorics of simplicial complexes in terms of the bigraded cohomology rings of moment-angle complexes. In the case when K is a triangulated manifold, the important additional information is provided by the bigraded Poincare duality for the Lefschetz pair (ZK ; Z? ). For instance, the duality implies the generalized Dehn{Sommerville equations for the numbers of k-simplices in a triangulated manifold.


INTRODUCTION

3

Each chapter and most sections of the book refer to a separate sub ject and contain necessary introductory remarks. Below we schematically overview the contents. The chapter dependence chart is shown in Figure 0.1.
?@ ? @ -4 ?
3

1

HH HH HH

- 2?

?

?

@

@ R @ -6 ? ?

-7

-8

HH ? HH ? j H5

Figure 0.1. Chapter dependence scheme.

Chapter 1 contains combinatorial and geometrical background material on convex polytopes. Since a lot of literature is available on this sub ject (see e.g. recent excellent lectures 145] by G. Ziegler), we just give a short overview of constructions used in the book. Although most of these constructions descend from the convex geometry, we tried to emphasize their combinatorial properties. Section 1.1 contains two classical de nitions of convex polytopes, examples, the notions of simple and simplicial polytopes, and the construction of connected sum of simple polytopes. In section 1.2 we introduce the f - and the h-vector of a polytope and give a \Morse-theoretical" proof of the Dehn{Sommerville equations. Section 1.3 is devoted to the g-theorem, and in section 1.4 we discuss the Upper Bound and the Lower Bound for the number of faces of a simple (or simplicial) polytope. In section 1.5 we introduce the Stanley{Reisner ring of a simple polytope. Simplicial complexes appear in the full generality in Chapter 2. In section 2.1 we de ne abstract and geometrical simplicial complexes (polyhedrons). In section 2.2 we introduce some standard notions from P L-topology and describe basic constructions of simplicial complexes (joins, connected sums etc.). We also discuss the Alexander duality and its simplicial version here. From the early days of topology, triangulations of nice topological spaces such as manifolds or spheres were of particular interest. Triangulations of spheres, or \simplicial spheres", are the sub ject of section 2.3. Here we also discuss the inter-relations between some particular subclasses of simplicial spheres (such as P L-spheres, polytopal spheres etc.) and one famous combinatorial problem, the so-called g-conjecture for face vectors. Triangulated (or simplicial) manifolds are the sub ject of section 2.4; some related open problems from low-dimensional and P L topology are also included there. The notion of bistellar moves, as a particularly interesting and useful class of operations on simplicial complexes, is discussed in section 2.5. In chapter 3 we give an overview of commutative algebra involved in the combinatorics of simplicial complexes. Many of the constructions from this chapter, especially those appearing in the beginning, are taken from Stanley's monograph 128]; however, we tried to emphasize their functorial properties and relationships with operations from chapter 2. The Stanley{Reisner face ring of simplicial complex is introduced in section 3.1. The important class of Cohen{Macaulay complexes is the


4

INTRODUCTION

sub ject of section 3.2; we also give Stanley's argument for the Upper Bound theorem for spheres here. Section 3.3 contains the homological algebra background, including resolutions and the graded functor Tor. Koszul complexes and Tor-algebras associated with simplicial complexes are described in section 3.4 together with their basic properties. Gorenstein algebras and Gorenstein* complexes are the sub ject of section 3.5. This class of \self-dual" Cohen{Macaulay complexes contains simplicial and homology spheres and, in a sense, provides the best possible algebraic approximation to them. The chapter ends up with a discussion of some generalizations of the Dehn{Sommerville equations. Cubical complexes are the sub ject of chapter 4. We give de nitions and discuss some interesting related problems from the discrete geometry in section 4.1. Section 4.2 introduces some particular cubical complexes necessary for the construction of moment-angle complexes. These include the cubical subdivisions of simple polytopes and simplicial complexes. Di erent aspects of torus actions is the main theme of the second part of the book. Chapter 5 starts with a brief review of the algebraic geometry of toric varieties in section 5.1. We stress upon those features of toric varieties which can be taken as a starting point for their subsequent topological generalizations. We also give Stanley's famous argument for the necessity part of the g-theorem, one of the rst and most known applications of the algebraic geometry in the combinatorics of polytopes. In section 5.2 we give the de nition and basic properties of quasitoric manifolds, the notion introduced by Davis and Januszkiewicz (under the name \toric manifolds") as a topological generalization of toric varieties. The topology of quasitoric manifolds is the sub ject of sections 5.3 and 5.4 (this includes the discussion of their cohomology, cobordisms, characteristic classes, Hirzebruch genera, etc.). Quasitoric manifolds work particularly well in the cobordism theory and may serve as a convenient framework for di erent cobordism calculations. Evidences for this are provided by some recent results of V. Buchstaber and N. Ray. It is proved that a certain class of quasitoric manifolds provides an alternative additive basis for the complex cobordism ring. (Note that the standard basis consists of Milnor hypersurfaces, which are not quasitoric.) Moreover, using the combinatorial construction of connected sum of polytopes, it is proved that each complex cobordism class contains a quasitoric manifold with a canonical stably almost complex structure respected by the torus action. Since quasitoric manifolds are necessarily connected, the nature of this result resembles the famous Hirzebruch problem about connected algebraic representatives in complex cobordisms. All these arguments, presented in section 5.3, open the way to evaluation of global cobordism invariants on manifolds by choosing a quasitoric representative and studying the local invariants of the action. As an application, in section 5.4 we give combinatorial formulae, due to the second author, for Hirzebruch genera of quasitoric manifolds. Section 5.5 is a discussion of several known results on the classi cation of toric and quasitoric manifolds over a given simple polytope. The theory of moment-angle complexes is the sub ject of chapters 6 and 7. We start in section 6.1 with the de nition of the moment-angle manifold ZP corresponding to a simple polytope P . The general moment-angle complexes ZK are introduced in section 6.2, using special cubical subdivisions from section 4.2. Here we prove that ZK is a manifold provided that K is a simplicial sphere. Two types of bigraded cell decompositions of moment-angle complexes are introduced in section 6.3. In section 6.4 we discuss di erent functorial properties of moment-angle


INTRODUCTION

5

complexes with respect to simplicial maps and constructions from section 2.2. A basic homotopy theory of moment-angle complexes is the sub ject of section 6.5. Concluding section 6.6 aims for a more broad view on the constructions of quasitoric manifolds and moment-angle complexes. We discuss di erent inter-relations, similar constructions and possible generalizations there. The cohomology of moment-angle complexes, and its r^ in investigating comole binatorial invariants of triangulations, is studied in chapter 7. In section 7.1 we review the Eilenberg{Moore spectral sequence, our main computational tool. The bigraded cellular structure and the Eilenberg{Moore spectral sequence are the main ingredients in the calculation of cohomology of a general moment-angle complex ZK , carried out in section 7.2. Additional results on the cohomology in the case when K is a simplicial sphere are given in section 7.4. These calculations reveal some new connections with well-known constructions from homological algebra and open the way to some further combinatorial applications. In particular, the cohomology of the Koszul complex for a Stanley{Reisner ring and its Betti numbers now get a topological interpretation. In section 7.5 we study the quotients of momentangle manifolds ZP by subtori H T m of rank < m. Quasitoric manifolds arise in this scheme as quotients for freely acting subtori of the maximal possible rank. Moment-angle complexes corresponding to triangulated manifolds are considered in section 7.6. In this situation all singular points of ZK form a single orbit of the torus action, and the complement of an equivariant neighborhood of this orbit is a manifold with boundary. In chapter 8 we apply the theory of moment-angle complexes to the topology of subspace arrangement complements. Section 8.1 is a brief review of general arrangements. Then we restrict to the cases of coordinate subspace arrangements and diagonal subspace arrangements (sections 8.2 and 8.3 respectively). In particular, we calculate the cohomology ring of the complement of a coordinate subspace arrangement by reducing it to the cohomology of a moment-angle complex. This also reveals some remarkable connections between certain results from commutative algebra of monomial ideals (such as the famous Hochster's theorem) and topological results on subspace arrangements (e.g. the Goresky{Macpherson formula for the cohomology of complement). In the diagonal subspace arrangement case the cohomology of complement is included as a canonical subspace into the cohomology of the loop space on a certain moment angle complex ZK . Almost all new concepts in our book are accompanied with explanatory examples. We also give many examples of particular computations, illustrating general theorems. Throughout the text the reader will encounter a number of open problems. Some of these problems and conjectures are widely known, while others are new. In most cases we tried to give a topological interpretation for the question under consideration, which might provide an alternative approach to its solution. Many of those results in the book which are due to the authors have already appeared in their papers 30]{ 34], 111], 112], or papers 37], 38] by N. Ray and the rst author. We sometimes omit the corresponding quotations in the text. The whole book has grown up from our survey article 35].
Acknowledgements. Both authors are indebted to Sergey Novikov, whose in uence on our topological education cannot be overestimated. The rst author takes this opportunity to express special thanks to Nigel Ray for the very pleasant joint work during the last ten years, which in particular generated some of the


6

INTRODUCTION

ideas developed in this book. The second author also wishes to express his deep gratitude to N. Ray for extremely fruitful collaboration and sincere hospitality during his stay in Manchester. The authors wish to thank Levan Alania, Yusuf Civan, Natalia Dobrinskaya, Nikolai Dolbilin, Mikhail Farber, Konstantin Feldman, Ivan Izmestiev, Frank Lutz, Oleg Musin, Andrew Ranicki, Elmer Rees, Mikhail Shtan'ko, Mikhail Shtogrin, Vladimir Smirnov, James Stashe , Neil Strickland, Sergey Tarasov, Victor Vassiliev, Volkmar Welker, Sergey Yuzvinsky and Gunter Ziegler for the insight gained from discussions on the sub ject of our book. We are also thankful to all participants of the seminar \Topology and Computational Geometry", which is being held by O. R. Musin and the authors at the Department of Mathematics and Mechanics, Lomonosov Moscow State University. We would like to gratefully acknowledge the most helpful comments, corrections, and additional references suggested by the referees. Thanks to them, the text has been signi cantly enhanced in many places. The work of both authors was partially supported by the Russian Foundation for Fundamental Research, grant no. 99-01-00090, and the Russian Leading Scienti c School Support, grant no. 00-15-96011. The second author was also supported by the British Royal Society/NATO Postdoctoral Fellowship while visiting the Department of Mathematics at the University of Manchester.


CHAPTER 1

Polytopes
Both combinatorial and geometrical aspects of the theory of convex polytopes are exposed in a vast number of textbooks, monographs and papers. Among them are the classical monograph 69] by Grunbaum and more recent Ziegler's lectures 145]. Face vectors and other combinatorial questions are discussed in books by McMullen{Shephard 99], Br nsted 29], Yemelichev{Kovalev{Kravtsov 141] and survey article 87] by Klee and Kleinschmidt. These sources contain a host of further references. In this section we review some basic concepts and constructions used in the rest of the book. There are two algorithmically di erent ways to de ne a convex polytope in n-dimensional a ne Euclidean space Rn . Definition 1.1. A convex polytope is the convex hull of a nite set of points in some Rn . Definition 1.2. A convex polyhedron P is an intersection of nitely many half-spaces in some Rn : (1.1) P = x 2 Rn : hl i ; x i > ?ai ; i = 1; : : : ; m ; where l i 2 (Rn ) are some linear functions and ai 2 R, i = 1; : : : ; m. A (convex) polytope is a bounded convex polyhedron. Nevertheless, the above two de nitions produce the same geometrical ob ject, i.e. the subset of Rn is a convex hull of a nite point set if and only if it is a bounded intersection of nitely many half-spaces. This classical fact is proved in many textbooks on polytopes and convex geometry, see e.g. 145, Theorem 1.1]. Definition 1.3. The dimension of a polytope is the dimension of its a ne hull. Unless otherwise stated we assume that any n-dimensional polytope, or simply n-polytope , P n is a subset in n-dimensional ambient space Rn . A supporting hyperplane of P n is an a ne hyperplane H which intersects P n and for which the polytope is contained in one of the two closed half-spaces determined by the hyperplane. The intersection P n \ H is then called a face of the polytope. We also regard the polytope P n itself as a face; other faces are called proper faces . The boundary @ P n is the union of all proper faces of P n . Each face of an n-polytope is itself a polytope of dimension 6 n. 0-dimensional faces are called vertices , 1-dimensional faces are edges , and codimension one faces are facets . Two polytopes P1 Rn1 and P2 Rn2 of the same dimension are said to be a nely equivalent (or a nely isomorphic ) if there is an a ne map Rn1 ! Rn2 that is a bijection between the points of the two polytopes. Two polytopes are combinatorial ly equivalent if there is a bijection between their sets of faces that preserves the inclusion relation.
7

1.1. De nitions and main constructions


8

1. POLYTOPES

Note that two a nely isomorphic polytopes are combinatorially equivalent, but the opposite is not true. A more consistent de nition of combinatorial equivalence uses the combinatorial notions of poset and lattice. Definition 1.4. A poset (or nite partially ordered set) (S ; 6) is a nite set S equipped with a relation \6" which is re exive (x 6 x for all x 2 S ), transitive (x 6 y and y 6 z imply x 6 z ), and antisymmetric (x 6 y and y 6 x imply x = y). When the partial order is clear we denote the poset by just S . A chain in S is a totally ordered subset of S . Definition 1.5. The faces of a polytope P of all dimensions form a poset with respect to inclusion, called the face poset . Now we observe that two polytopes are combinatorially equivalent if and only if their face posets are isomorphic. More information about face posets of polytopes can be found in 145, x2.2]. Definition 1.6. A combinatorial polytope is a class of combinatorial equivalent convex (or geometrical ) polytopes. Equivalently, a combinatorial polytope is the face poset of a geometrical polytope. Agreement. Suppose that a polytope P n is represented as an intersection of half-spaces as in (1.1). In the sequel we assume that there are no redundant inequalities hl i ; x i > ?ai in such a representation. That is, no inequality can be removed from (1.1) without changing the polytope P n . In this case P n has exactly m facets which are the intersections of hyperplanes hl i ; x i = ?ai , i = 1; : : : ; m, with P n . The vector l i is orthogonal to the corresponding facet and points towards the interior of the polytope. Example 1.7 (simplex and cube). An n-dimensional simplex n is the convex hull of (n + 1) points in Rn that do not lie on a common a ne hyperplane. All faces of an n-simplex are simplices of dimension 6 n. Any two n-simplices are a nely equivalent. The standard n-simplex is the convex hull of points (1; 0; : : : ; 0), (0; 1; : : : ; 0); : : : ; (0; : : : ; 0; 1), and (0; : : : ; 0) in Rn . Alternatively, the standard nsimplex is de ned by (n + 1) inequalities (1.2) xi > 0; i = 1; : : : ; n; and ? x1 ? : : : ? xn > ?1: The regular n-simplex is the convex hull of n + 1 points (1; 0; : : : ; 0), (0; 1; : : : ; 0), : : : , (0; : : : ; 0; 1) in Rn+1 . The standard q-cube is the convex polytope I q Rq de ned by (1.3) I q = f(y1 ; : : : ; yq ) 2 Rq : 0 6 yi 6 1; i = 1; : : : ; qg: Alternatively, the standard q-cube is the convex hull of the 2q points in Rq that have only zero or unit coordinates. The following construction shows that any convex n-polytope with m facets is a nely equivalent to the intersection of the positive cone (1.4) Rm = (y1 ; : : : ; ym) 2 Rm : yi > 0; i = 1; : : : ; m Rm + with a certain n-dimensional plane. Construction 1.8. Let P Rn be a convex n-polytope given by (1.1) with some l i 2 (Rn ) , ai 2 R, i = 1; : : : ; m. Form the n m-matrix L whose columns are the vectors l i written in the standard basis of (Rn ) , i.e. Lji = (l i )j . Note that


1.1. DEFINITIONS AND MAIN CONSTRUCTIONS

9

Consider the a ne map (1.6) AP : Rn ! Rm ; AP (x ) = Ltx + a 2 Rm : Its image is an n-dimensional plane in Rm , and AP (P ) is the intersection of this plane with the positive cone Rm , see (1.5). Let W be an m (m ? n)-matrix whose + columns form a basis of linear dependencies between the vectors l i . That is, W is a rank (m ? n) matrix satisfying LW = 0. Then it is easy to see that AP (P ) = y 2 Rm : W t y = W t a ; yi > 0; i = 1; : : : ; m : By de nition, the polytopes P and AP (P ) are a nely equivalent. Example 1.9. Consider the standard n-simplex n Rn de ned by inequalities (1.2). It has m = n + 1 facets and is given by (1.1) with l 1 = (1; 0; : : : ; 0)t , : : : , l n = (0; : : : ; 0; 1)t , l n+1 = (?1; : : : ; ?1)t , a1 = : : : = an = 0, an+1 = 1. One can take W = (1; : : : ; 1)t in Construction 1.8. Hence, W t y = y1 + : : : + ym , W t a = 1, and we have A n ( n ) = y 2 Rn+1 : y1 + : : : + yn+1 = 1; yi > 0; i = 1; : : : ; n : This is the regular n-simplex in Rn+1 . The notion of generic polytope depends on the choice of de nition of convex polytope. Below we describe the two possibilities. A set of m > n points in Rn is in general position if no (n + 1) of them lie on a common a ne hyperplane. Now De nition 1.1 implies that a convex polytope is generic if it is the convex hull of a set of general positioned points. This implies that all proper faces of the polytope are simplices, i.e. every facet has the minimal number of vertices (namely, n). Such polytopes are called simplicial . On the other hand, a set of m > n hyperplanes hl i ; x i = ?ai , l i 2 (Rn ) , x 2 Rn , ai 2 R, i = 1; : : : ; m, is in general position if no point belongs to more than n hyperplanes. From the viewpoint of De nition 1.2, a convex polytope P n is generic if its bounding hyperplanes (see (1.1)) are in general position. That is, there are exactly n facets meeting at each vertex of P n . Such polytopes are called simple . Note that each face of a simple polytope is again a simple polytope. Definition 1.10. For any convex polytope P Rn de ne its polar set P n ) by (R P = fx 0 2 (Rn ) : hx 0 ; x i > ?1 for all x 2 P g: Remark. We adopt the de nition of the polar set from the algebraic geometry of toric varieties, not the classical one from the convex geometry. The latter is obtained by replacing the inequality \> ?1" above by \6 1". Obviously, the toric geometers polar set is taken into the convex geometers one by the central symmetry with respect to 0. It is well known in convex geometry that the polar set P is convex in the dual space (Rn ) and 0 is contained in the interior of P . Moreover, if P itself contains 0 in its interior then P is a convex polytope (i.e. is bounded) and (P ) = P ,

L is of rank n. Likewise, let a = (a1 ; : : entries ai . Then we can rewrite (1.1) as (1.5) P = x 2 Rn : (Lt x + where Lt is the transposed matrix and

: ; am )t 2 Rm be the column vector with
a )i > 0; i = 1; : : : ; m ; x = (x1 ; : : : ; xn )t is the column vector.


10

1. POLYTOPES

see e.g. 145, x2.3]. The polytope P is called the polar (or dual ) of P . There is a one-to-one order reversing correspondence between the face posets of P and P . In other words, the face poset of P is the opposite of the face poset of P . In particular, if P is simple then P is simplicial, and vice versa. Example 1.11. Any polygon (2-polytope) is simple and simplicial at the same time. In dimensions > 3 the simplex is the only polytope that is simultaneously simple and simplicial. The cube is a simple polytope. The polar of simplex is again the simplex. The polar of cube is called the cross-polytope . The 3-dimensional cross-polytope is known as the octahedron. Construction 1.12 (Product of simple polytopes). The product P1 P2 of two simple polytopes P1 and P2 is a simple polytope as well. The dual operation on simplicial polytopes can be described as follows. Let S1 Rn1 and S2 Rn2 be two simplicial polytopes. Suppose that both S1 and S2 contain 0 in their interiors. Now de ne ? S1 S2 := conv S1 0 0 S2 Rn1 +n2 (here conv means the convex hull). It is easy to see that S1 S2 is a simplicial polytope, and for any two simple polytopes P1 , P2 containing 0 in their interiors the following holds: P1 P2 = (P1 P2 ) : Obviously, both product and operations are also de ned on combinatorial polytopes; in this case the above formula holds without any restrictions. Construction 1.13 (Connected sum of simple polytopes). Suppose we are given two simple polytopes P n and Qn , both of dimension n, with distinguished vertices v and w respectively. The informal way to get the connected sum P n #v;w Qn of P n at v and Qn at w is as follows. We \cut o " v from P n and w from Qn; then, after a pro jective transformation, we can \glue" the rest of P n to the rest of Qn along the new simplex facets to obtain P n #v;w Qn . Below we give the formal de nition, following 38, x6]. First, we introduce an n-polyhedron ? n , which will be used as a template for the construction; it arises by considering the standard (n ? 1)-simplex n?1 in the subspace fx : x1 = 0g of Rn , and taking its cartesian product with the rst coordinate axis. The facets Gr of ? n therefore have the form R Dr , where Dr , 1 r n, are the facets of n?1 . Both ? n and the Gr are divided into positive and negative halves, determined by the sign of the coordinate x1 . We order the facets of P n meeting in v as E1 ; : : : ; En , and the facets of Qn meeting in w as F1 ; : : : ; Fn . Denote the complementary sets of facets by Cv and Cw ; those in Cv avoid v, and those in Cw avoid w. We now choose pro jective transformations P and Q of Rn , whose purpose is to map v and w to x1 = 1 respectively. We insist that P embeds P n in ? n so as to satisfy two conditions; rstly, that the hyperplane de ning Er is identi ed with the hyperplane de ning Gr , for each 1 r n, and secondly, that the images of the hyperplanes de ning Cv meet ? n in its negative half. Similarly, Q identi es the hyperplane de ning Fr with that de ning Gr , for each 1 r n, but the images of the hyperplanes de ning Cw meet ? n in its positive half. We de ne the connected sum P n #v;w Qn of P n at v and Qn at w to be the simple convex n-polytope determined by the images of the hyperplanes de ning Cv and Cw and hyperplanes de ning Gr , r = 1; : : : ; n. It is de ned only up to combinatorial equivalence;


1.2. FACE VECTORS AND DEHN{SOMMERVILLE EQUATIONS

11

moreover, di erent choices for either of v and w, or either of the orderings for Er and Fr , are likely to a ect the combinatorial type. When the choices are clear, or their e ect on the result irrelevant, we use the abbreviation P n # Qn . The related construction of connected sum P # S of a simple polytope P and a simplicial polytope S is described in 145, Example 8.41]. Example 1.14. 1. If P 2 is an m1 -gon and Q2 is an m2 -gon then P # Q is an (m1 + m2 ? 2)-gon. 2. If both P and Q are n-simplices then P # Q = n?1 I 1 (the product of (n ? 1)-simplex and segment). In particular, for n = 3 we get a triangular prism. 3. More generally, if P is an n-simplex then P #v;w Q is the result of \cutting" the vertex w from Q by a hyperplane that isolate w from other vertices. For more relations between connected sums and hyperplane cuts see 38, x6]. Definition 1.15. A simplicial polytope S is called k-neighborly if any k vertices span a face of S . Likewise, a simple polytope P is called k-neighborly if any k facets of P have non-empty intersection (i.e. share a common codimension-k face). Obviously, every simplicial (or simple) polytope is 1-neighborly. It can be shown ( 29, Corollary 14.5], see also Example 1.24 below) that if S is a k-neighborly simplicial n-polytope and k > n , then S is an n-simplex. This implies that any 2 2-neighborly simplicial 3-polytope is a simplex. However, there exist simplicial n-polytopes with an arbitrary number of vertices which are n -neighborly. Such 2 polytopes are called neighborly . In particular, there is a simplicial 4-polytope (different from the 4-simplex) any two vertices of which are connected by an edge. 2 be the product of Example 1.16 (neighborly 4-polytope). Let P = 2 two triangles. Then P is a simple polytope, and it is easy to see that any two facets of P share a common 2-face. Hence, P is 2-neighborly. The polar P is a neighborly simplicial 4-polytope. More generally, if a simple polytope P1 is k1 -neighborly and a simple polytope P2 is k2 -neighborly, then the product P1 P2 is a min(k1 ; k2 )-neighborly simple n ) and ( n n+1 ) provide examples of polytope. It follows that ( n neighborly simplicial 2n- and (2n + 1)-polytopes. The following example gives a neighborly polytope with an arbitrary number of vertices. Example 1.17 (cyclic polytopes). De ne the moment curve in Rn by x : R ?! Rn ; t 7! x (t) = (t; t2 ; : : : ; tn ) 2 Rn : For any m > n de ne the cyclic polytope C n (t1 ; : : : ; tm ) as the convex hull of m distinct points x (ti ), t1 < t2 < : : : < tm , on the moment curve. It follows from the Vandermonde determinant identity that no (n + 1) points on the moment curve belong to a common a ne hyperplane. Hence, C n (t1 ; : : : ; tm ) is a simplicial npolytope. It can be shown (see 145, Theorem 0.7]) that C n (t1 ; : : : ; tm ) has exactly m vertices x (ti ), the combinatorial type of cyclic polytope does not depend on the speci c choice of the parameters t1 ; : : : ; tm , and C n (t1 ; : : : ; tm ) is a neighborly simplicial n-polytope. We will denote the combinatorial cyclic n-polytope with m vertices by C n (m). The notion of the f -vector (or face vector) is a central concept in the combinatorial theory of polytopes. It has been studied there since the time of Euler.

1.2. Face vectors and Dehn{Sommerville equations


12

1. POLYTOPES

Definition 1.18. Let S be a simplicial n-polytope. Denote by fi the number of i-dimensional faces of S . The integer vector f (S ) = (f0 ; : : : ; fn?1 ) is called the f -vector of S . We also put f?1 = 1. The h-vector of S is the integer vector (h0 ; h1 ; : : : ; hn ) de ned from the equation (1.7) h0 tn + : : : + hn?1 t + hn = (t ? 1)n + f0 (t ? 1)n?1 + : : : + fn?1 : Finally, the sequence (g0 ; g1 ; : : : ; g n ) where g0 = 1, gi = hi ? hi?1 , i > 0, is called 2 the g-vector of S . The f -vector of a simple n-polytope P n is de ned as the f -vector of its polar: f (P ) := f (P ), and similarly for the h- and the g -vector of P . More explicitly, f (P ) = (f0 ; : : : ; fn?1 ), where fi is the number of faces of P of codimension (i + 1) (i.e. of dimension (n ? i ? 1)). In particular, f0 is the number of facets of P , which we usually denote m(P ) or just m. The agreement f?1 = 1 is now justi ed by the fact that P itself is a face of codimension 0. Remark. The de nition of h-vector may seem to be unnatural at rst glance. However, as we will see later, the h-vector has a number of combinatorial-geometrical and algebraic interpretations and in some situations is more convenient to work with than the f -vector. Obviously, the f -vector is a combinatorial invariant of P n , that is, it depends only on the face poset. For convenience we assume all polytopes in this section to be combinatorial, unless otherwise stated. Example 1.19. Two di erent combinatorial simple polytopes may have same f -vectors. For instance, let P13 be the 3-cube and P23 be the simple 3-polytope with 2 triangular, 2 quadrangular and 2 pentagonal facets, see Figure 1.1. (Note that P23 is dual to the cyclic polytope C 3 (6) from De nition 1.17.) Then f (P13 ) = f (P23 ) = (6; 12; 8).
H HH HH H H H H H ? PP @ PP P @ @

Z Z Z @ @

Z

@

Z Z @

@

@

Z? Z Z Z ? Z @ Z @ @ @ @ @

Figure 1.1. Two combinatorially non-equivalent simple polytopes with the same f -vectors.

The f -vector and the h-vector carry the same information about the polytope and determine each other by means of linear relations, namely (1.8)

hk =

k X i=0

? n?i (?1)k?i n?k fi?1 ; fn?1?k =

n X? q=k

q k hn?q

; k = 0; : : : ; n:


1.2. FACE VECTORS AND DEHN{SOMMERVILLE EQUATIONS
?

13

In particular, h0 = 1 and hn = (?1)n 1 ? f0 + f1 + : : : + (?1)n fn?1 . By Euler's theorem, (1.9) f0 ? f1 + + (?1)n?1 fn?1 = 1 + (?1)n?1 ; which is equivalent to hn = h0 (= 1). In the case of simple polytopes Euler's theorem admits the following generalization. Theorem 1.20 (Dehn{Sommerville relations). The h-vector of any simple or simplicial n-polytope is symmetric, i.e. hi = hn?i ; i = 0; 1; : : : ; n: The Dehn{Sommerville equations can be proved in many di erent ways. We give a proof which uses a Morse-theoretical argument, which rst appeared in 29]. We will return to this argument in chapter 5. Proof of Theorem 1.20. Let P n Rn be a simple polytope. Choose a linear function ' : Rn ! R which is generic in the sense that it distinguishes the vertices of P n . For this ' there is a vector in Rn such that '(x ) = h ; x i. The assumption on ' implies that is parallel to no edge of P n . Now we can view ' as a height function on P n . Using ', we make the 1-skeleton of P n a directed graph by orienting each edge in such a way that ' increases along it (this can be done since ' is generic), see Figure 1.2. For each vertex v of P n de ne its index,
y X 3X XXX ] J XXX J XXX ind = 3 J X J M B J B J B J B I @ @ J B @ JX B yX XX @ ? XXX B @ ? ind = 2 XX B @ ? @ ? K A A @ ? A @? A M B ind = 1 B A B AH Y HH B HH B HH B HH B : HH B ind = 0 H B

6

Figure 1.2. Oriented 1-skeleton of P and index of vertex.

ind(v), as the number of incident edges that point towards v. Denote the number of vertices of index i by I (i). We claim that I (i) = hn?i . Indeed, each face of P n has a unique top vertex (the maximum of the height function ' restricted to the face) and a unique bottom vertex (the minimum of '). Let F k be a k-face of P n , and vF its top vertex. Since P n is simple, there are exactly k edges of F k meeting


14

1. POLYTOPES

at vF , whence ind(vF )? > k. On the other hand, each vertex of index q > k is the q top vertex for exactly k faces of dimension k. It follows that fn?1?k (the number of k-faces) can be calculated as X? q fn?1?k = k I (q ): Now, the second identity from (1.8) shows that I (q) = hn?q , as claimed. In particular, the number I (q) does not depend on . At the same time, since ind (v) = n ? ind? (v) for any vertex v, one has hn?q = I (q) = I? (n ? q) = hq :
q>k

f -vector as follows:
(1.10)

Using (1.8), we can rewrite the Dehn{Sommerville equations in terms of the

fk?1 =

n X j =k

?j (?1)n?j k fj?1 ; k = 0; 1; : : : ; n:

The Dehn{Sommerville equations were established by Dehn 52] for n 6 5 in 1905, and by Sommerville in the general case in 1927 122] in the form similar to (1.10). n n Example 1.21. Let P1 1 and P2 2 be simple polytopes. Each face of P1 P2 is the product of a face of P1 and a face of P2 , whence

fk (P1 P2 ) =
Set

n1 ?1 X

i=?1

fi (P1 )fk?i?1 (P2 ); k = ?1; 0; : : : ; n1 + n2 ? 1:

+ hn tn : Then it follows from the above formula and (1.7) that (1.11) h(P1 P2 ; t) = h(P1 ; t)h(P2 ; t): Example 1.22. Let us express the f -vector and the h-vector of the connected sum P n # Qn in terms of that of P n and Qn . From Construction 1.13 we deduce that ?n fi (P n # Qn) = fi (P n ) + fi (Qn ) ? i+1 ; i = 0; 1; : : : ; n ? 2; fn?1 (P n # Qn) = fn?1 (P n ) + fn?1 (Qn ) ? 2: Then it follows from (1.8) that h0 (P n # Qn ) = hn (P n # Qn ) = 1; hi (P n # Qn ) = hi (P n ) + hi (Qn ); i = 1; 2; : : : ; n ? 1: This property raises the following question. Problem 1.23. Describe al l integer-valued functions on the set of simple polytopes which are linear with respect to the connected sum operation. The previous identities show that examples of such functions are provided by hi for i = 1; : : : ; n ? 1.

h(P ; t) = h0 + h1 t +


1.3. THE g-THEOREM

15

Example 1.24. Suppose S is a q-neighborly simplicial n-polytope (see De ni? tion 1.15) di erent from the n-simplex. Then fk?1 (S ) = m , k 6 q. From (1.8) k we get

(1.12)

hk (S ) =

k X i=0

?? ? ? + (?1)k?i n?ii m = m?nk k?1 ; k 6 q: k i

The latter equality is obtained by calculating the coe cient of tk from two sides of the identity 1 m m?n+k?1 : (1 + t)n?k+1 (1 + t) = (1 + t) Since S is not a simplex, we have m > n + 1, which together with (1.12) gives h0 < h1 < < hq . These inequalities together with the Dehn{Sommerville equations imply the upper bound q 6 n mentioned in De nition 1.15. 2 The g-theorem gives answer to the following natural question: which integer vectors may appear as the f -vectors of simple (or, equivalently, simplicial) polytopes? The Dehn{Sommerville relations provide a necessary condition. As far as only linear equations are concerned, there are no further restrictions. Proposition 1.25 (Klee 86]). The Dehn{Sommervil le relations are the most general linear equations satis ed by the f -vectors of al l simple (or simplicial) polytopes. Proof. In 86] the statement was proved directly, in terms of f -vectors. However, the usage of h-vectors signi cantly simpli es the proof. It is su cient to prove that the a ne hull of the h-vectors (h0 ; h1 ; : : : ; hn ) of simple n-polytopes is an n 2 dimensional plane. This can be done, for instance, by providing n + 1 simple poly2 topes with a nely independent h-vectors. Set Qk := k n?k , k = 0; 1 : : : ; n . 2 Since h( k ) = 1 + t + + tk , the formula (1.11) gives n?k+1 tk+1 h(Qk ) = 1 ?? t 1 ?1t? t : 1 It follows that h(Qk+1 ) ? h(Qk ) = tk+1 + + tn?k?1 , k = 0; 1; : : : ; n ? 1. 2 Therefore, the vectors h (Qk ), k = 0; 1; : : : ; n , are a nely independent. 2

1.3. The g-theorem

n edges and each edge connects two vertices. This implies the following \obvious" linear equation for the components of the f -vector of P n : (1.13) 2fn?2 = nfn?1:
Proposition 1.25 shows that this equation is a corollary of the Dehn{Sommerville equations. (One may observe that it is equation (1.10) for k = n ? 1.) It follows from (1.13) and Euler identity (1.9) that for simple (or simplicial) 3-polytopes the f -vector is completely determined by the number of facets, namely, f (P 3 ) = (f0 ; 3f0 ? 6; 2f0 ? 4):

Example 1.26. Each vertex of a simple n-polytope P n is contained in exactly


16

1. POLYTOPES

We mention also that Euler identity (1.9) is the only linear relation satis ed by the face vectors of general convex polytopes. (This can be proved in a similar way as Proposition 1.25, by specifying su ciently many polytopes with a nely independent face vectors.) The conditions characterizing the f -vectors of simple (or simplicial) polytopes, now know as the g-theorem , were conjectured by P. McMullen 96] in 1970 and proved by R. Stanley 125] (necessity) and Billera and Lee 18] (su ciency) in 1980. Besides the Dehn{Sommerville equations, the g-theorem contains two groups of inequalities, one linear and one non-linear. To formulate the g-theorem we need the following construction. Definition 1.27. For any two positive integers a, i there exists a unique binomial i-expansion of a of the form ? ?? ? a = aii + aii?11 + + ajj ; where ai > ai?1 > > aj > j > 1. De ne ? +1 ? ? +1 ahii = aii+1 + ai?i1 +1 + + ajj+1 ; 0hii = 0: 2. If i > a then the binomial expansion has the form ? ? ? a = ii + ii?1 + + ii?a+1 = 1 + + 1; ?1 ?a+1 and therefore ahii = a. 3. Let a = 28, i = 4. The corresponding binomial expansion is ? ? ? 28 = 6 + 5 + 3 : 4 3 2 Hence, ? ? ? 28h4i = 7 + 6 + 4 = 40: 5 4 3 Theorem 1.29 (g-theorem). An integer vector (f0 ; f1 ; : : : ; fn?1 ) is the f -vector of a simple n-polytope if and only if the corresponding sequence (h0 ; : : : ; hn ) determined by (1.7) satis es the fol lowing three conditions: (a) hi = hn?i , i = 0; : : : ; n (the Dehn{Sommervil le equations); (b) h0 6 h1 6 : : : 6 h n , i = 0; 1; : : : ; n ; 2 2 (c) h0 = 1, hi+1 ? hi 6 (hi ? hi?1 )hii , i = 1; : : : ; n ? 1. 2 Remark. Obviously, the same conditions characterize the f -vectors of simplicial polytopes. Example 1.30. 1. The rst inequality h0 6 h1 from part (b) of g-theorem is equivalent to f0 = m > n + 1. This just expresses the fact that it takes at least n + 1 hyperplanes to bound a polytope in Rn . 2. Taking into account that ? h2 = n ? (n ? 1)f0 + f1 2 (see (1.8)), we rewrite the rst inequality h2 ? h1 6 (h1 ? h0 )h1i from part (c) of g-theorem as ?n+1 ?f ?n 0 2 ? nf0 + f1 6 2 : (see Example 1.28.1). This is equivalent to the upper bound
? f1 6 f20 ; ? Example 1.28. 1. For a > 0, ah1i = a+1 . 2


1.3. THE g-THEOREM

17

which says that two facets share at most one face of codimension two. In the dual \simplicial" notations, two vertices are joined by at most one edge. 3. The second inequality h1 6 h2 (for n > 4) from part (b) of g-theorem is equivalent to ? This is the rst (and most signi cant) inequality from the famous Lower Bound Conjecture for simple polytopes (see Theorem 1.37 below).

f1 > nf0 ? n
h1 (h1 +1)
2

+1 2

:

h

26

h2

=

? ?

?

?

?

?

?

?

?

?

?

? ?

? ?h2

=

h

1

0
Figure 1.3. (h1 ; h2 )-domain, n

-

h

1

> 4.

Thus, the rst two coordinates of the h-vectors of simple polytopes P n , n > 4, fall between the two curves h2 = h1 (h21 +1) and h2 = h1 in the (h1 ; h2 )-plane (see Figure 1.3). Note that the most general linear inequalities satis ed by points from this domain are h1 > 1 and h2 > h1 . Definition 1.31. An integral sequence (k0 ; k1 ; : : : ; kr ) satisfying k0 = 1 and 0 6 ki+1 6 kihii for i = 1; : : : ; r ? 1 is called an M -vector (after M. Macaulay). Conditions (b) and (c) from g-theorem are equivalent to that the g-vector (g0 ; g1; : : : ; g n ) of a simple n-polytope is an M -vector. The notion of M -vector 2 arises in the following classi cation result of commutative algebra. Theorem 1.32. An integral sequence (k0 ; k1 ; : : : ; kr ) is an M -vector if and A2r over a only if there exists a commutative graded algebra A = A0 A2 0 such that eld k = A (a) A is generated (as an algebra) by degree-two elements; (b) the dimension of 2i-th graded component of A equals ki : dimk A2i = ki ; i = 1; : : : ; r: This theorem is essentially due to Macaulay, but the above explicit formulation is that of 124]. The proof can be also found there. The proof of the su ciency part of g-theorem, due to Billera and Lee, is quite elementary and relies upon a remarkable combinatorial-geometrical construction of a simplicial polytope with any prescribed M -sequence as its g-vector. On the other


18

1. POLYTOPES

hand, Stanley's proof of the necessity part of g-theorem (i.e. that the g-vector of a simple polytope is an M -vector) used deep results from algebraic geometry, in particular, the Hard Lefschetz theorem for the cohomology of toric varieties. We outline Stanley's proof in section 5.1. After 1993 several more elementary combinatorial proofs of the g-theorem appeared. The rst such proof is due to McMullen 97]. It builds up on the notion of polytope algebra , which substitutes the cohomology algebra of toric variety. Despite being elementary, it was a complicated proof. Later, McMullen simpli ed his approach in 98]. Yet another elementary proof of the g-theorem has been recently found by Timorin 133]. It relies on the interpretation of McMullen's polytope algebra as the algebra of di erential operators (with constant coe cients) vanishing on the volume polynomial of the polytope. The following statement, now know as the Upper Bound Conjecture (UBC), was suggested by Motzkin in 1957 and proved by P. McMullen 95] in 1970. Theorem 1.33 (UBC for simplicial polytopes). From al l simplicial n-polytopes S with m vertices the cyclic polytope C n (m) (Example 1.17) has the maximal number of i-faces, 2 6 i 6 n ? 1. That is, if f0 (S ) = m, then ? fi (S ) 6 fi C n (m) for i = 2; : : : ; n ? 1: The equality in the above formula holds if and only if S is a neighborly polytope (see De nition 1.15). Note that since C n (m) is neighborly, ? ?m fi C n (m) = i+1 for i = 0; : : : ; n ? 1: 2 Due to the Dehn{Sommerville equations this determines the full f -vector of C n (m). The exact values are given by the following lemma. Lemma 1.34. The number of i-faces of cyclic polytope C n (m) (or any neighborly n-polytope with m vertices) is given by

1.4. Upper Bound and Lower Bound theorems

fi C n (m) =

?

2 X?

n

where we assume p = 0 for p < q. q Proof. Using the second identity from (1.8), identity n + 1 = n ? n?1 , the 2 2 Dehn{Sommerville equations, and (1.12), we calculate

q=0

m?n+q?1 + n?1?i q q

?

n?1 p=0

2 X?

n?p ?m?n+p?1 i+1?p p

; i = ?1; : : : ; n?1;

?

fi =
=

n X? q=0 n

n?1?i hn?q = q
?

q

2 X?

n

q=0

n?1?i hq + n?1 p=0
2 X?

q

n X q= n +1 2

?

q n?1?i hn?q

2 X?

q=0

m?n+q?1 + n?1?i q

n?p ?m?n+p?1 i+1?p p

:

The above proof justi es the following statement.


1.4. UPPER BOUND AND LOWER BOUND THEOREMS

19

Corollary 1.35. The UBC for simplicial polytopes (Theorem 1.33) is implied by the fol lowing inequalities for the h-vector of a simplicial polytope S with m vertices ? hi (S ) 6 m?ni+i?1 ; i = 0; : : : ; n : 2 This was one of the key observations in McMullen's original proof of the UBC for simplicial polytopes (see also 29, x18] and 145, x8.4]). The above corollary is also useful for a di erent generalization of UBC (we will return to this in section 3.2). We note also that due to the argument of Klee and McMullen (see 145, Lemma 8.24]) the UBT holds for al l convex polytopes, not necessarily simplicial. That is, the cyclic polytope C n (m) has the maximal number of i-faces from all convex n-polytopes with m vertices. Another fundamental fact from the theory of convex polytopes is the Lower Bound Conjecture (LBC) for simplicial polytopes. Definition 1.36. A simplicial n-polytope S is called stacked if there is a sequence S0 ; S1 ; : : : ; Sk = S of n-polytopes such that S0 is an n-simplex and Si+1 is obtained from Si by adding a pyramid over some facet of Si . In the combinatorial language, stacked polytopes are those obtained from a simplex by applying several subsequent stellar subdivisions of facets. Remark. Adding a pyramid (or stellar subdivision of a facet) is dual to \cutting a vertex" of a simple polytope (see Example 1.14.3). Theorem 1.37 (LBC for simplicial polytopes). For any simplicial n-polytope S (n > 3) with m = f0 vertices hold ? ? +1 fi (S ) > n f0 ? n+1 i for i = 1; : : : ; n ? 2; i i fn?1 (S ) > (n ? 1)f0 ? (n + 1)(n ? 2): The equality is achieved if and only if S is a stacked polytope. The argument by McMullen, Perles and Walkup 100] reduces the LBC to the ? case i = 1, namely, the inequality f1 > f0 ? n+1 . The LBC was rst proved by 2 Barnette 13], 15]. The \only if " part in the statement about the equality was proved in 19] using g-theorem. Unlike the UBC, little is know about generalizations of the LBC theorem to non-simplicial convex polytopes. Some results in this direction were obtained in 83] along with generalizations of the LBC theorem to simplicial spheres and manifolds (see also sections 2.3{2.4 in this book). In dual notations, the UBC and the LBC provide upper and lower bounds for the number of faces of simple polytopes with given number of facets. Both theorems were proved approximately at the same time (in 1970) and motivated P. McMullen to conjecture the g-theorem 96]. On the other hand, both UBC and LBC are corollaries of the g-theorem (see e.g. 29, x20]). In fact the LBC follows only from parts (a) and (b) of Theorem 1.29, while the UBC follows from parts (a) and (c). Part (b) of g-theorem, namely the inequalities h0 6 h1 6 : : : 6 h n ; (1.14)

where suggested in 100] as a generalization of the LBC for simplicial polytopes. The second inequality h1 6 h2 is equivalent to the i = 1 case of LBC (see Example 1.30.3). It follows from the results of 100] and 19] that (1.14) are the strongest

2


20

1. POLYTOPES

possible linear inequalities satis ed by the f -vectors of simple (or simplicial) polytopes (compare with the comment after Example 1.30). These inequalities are now known as the Generalized Lower Bound Conjecture (GLBC). During the last two decades a lot of work was done in extending the Dehn{ Sommerville equations, the GLBC and the g-theorem to ob jects more general than simplicial polytopes. However, there are still many intriguing open problems here. For more information see the rst section of survey article 129] by Stanley and section 2.3 in this book. The only aim of this short section is to de ne the Stanley{Reisner ring of a simple polytope. This fundamental combinatorial invariant will be one of the main characters in the next chapter. However, it is convenient for us to give it an independent treatment in the polytopal case. Let P be a simple n-polytope with m facets F1 ; : : : ; Fm . Fix a commutative ring k with unit. Let k v1 ; : : : ; vm] be the polynomial algebra over k on m generators. We make it a graded algebra by setting deg(vi ) = 2. Definition 1.38. The face ring (or the Stanley{Reisner ring ) of a simple polytope P is the quotient ring k(P ) = k v1 ; : : : ; vm]=IP ; where IP is the ideal generated by all square-free monomials vi1 vi2 vis such that Fi1 \ \ Fis = ? in P , i1 < < is . Since IP is a homogeneous ideal, k(P ) is a graded k-algebra. Remark. In certain circumstances it is convenient to choose a di erent grading in k v1 ; : : : ; vm ] and correspondingly k(K ). These cases will be particularly mentioned. Example 1.39. 1. Let P n be the n-simplex (regarded as a simple polytope). Then k(P n ) = k v1 ; : : : ; vn+1 ]=(v1v2 vn+1): 2. Let P be the 3-cube I 3 . Then k(P ) = k v1 ; v2 : : : ; v6 ]=(v1v4 ; v2v5 ; v3v6): 2 be the m-gon, m > 4. Then 3. Let P IP 2 = (vi vj : i ? j 6= 0; 1 mod m):

1.5. Stanley{Reisner face rings of simple polytopes


CHAPTER 2

Topology and combinatorics of simplicial complexes
Simplicial complexes or triangulations ( rst introduced by Poincare) provide an elegant, rigorous and convenient tool for studying topological invariants by combinatorial methods. The algebraic topology itself evolved from studying triangulations of topological spaces. With the appearance of cellular (or CW) complexes algebraic tools gradually replaced the combinatorial ones in topology. However, simplicial complexes have always played a signi cant role in P L topology, discrete and combinatorial geometry. The convex geometry provides an important class of sphere triangulations which are the boundary complexes of simplicial polytopes. The emergence of computers resulted in regaining the interest to \Combinatorial Topology", since simplicial complexes provide the most e ective way to translate topological structures into machine language. So, it seems to be the proper time for topologists to make use of remarkable achievements in discrete and combinatorial geometry of the last decades, which we started to review in the previous chapter.

of an abstract simplicial complex is the maximal dimension of its simplices. A simplicial complex K is pure if all its maximal simplices have the same dimension. A subcollection K 0 K which is also a simplicial complex is called a subcomplex of K . In most of our constructions it is safe to x an ordering in S and identify S with the index set m] = f1; : : : ; mg. This makes the notation more clear; however, in some cases it is more convenient to keep unordered sets. To distinguish from abstract simplices, the convex polytopes introduced in Example 1.7 (i.e. the convex hulls of a nely independent points) will be referred to as geometrical simplices. Definition 2.2. A geometrical simplicial complex (or a polyhedron ) is a subset P Rn represented as a nite union U of geometrical simplices of any dimensions in such a way that the following two conditions are satis ed: (a) each face of a simplex in U belongs to U ;
21

2.1. Abstract simplicial complexes and polyhedrons Let S be a nite set. Given a subset S , we denote its cardinality by j j. Definition 2.1. An (abstract) simplicial complex on the set S is a collection K = f g of subsets of S such that for each 2 K all subsets of (including ?) also belong to K . A subset 2 K is called an (abstract) simplex of K . One-element subsets are called vertices of K . If K contains all one-element subsets of S , then we say that K is a simplicial complex on the vertex set S . The dimension of an abstract simplex 2 K is its cardinality minus one: dim = j j ? 1. The dimension


22

2. TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

(b) the intersection of any two simplices in U is a face of each. A geometrical simplex from U is called a face of P ; as usual, one-dimensional faces are vertices . The dimension of P is the maximal dimension of its faces. Agreement. The notion of polyhedron from De nition 1.2 is not the same as that from De nition 2.2. The rst meaning of the term \polyhedron" (i.e. the \unbounded polytope") is adopted in the convex geometry, while the second one (i.e. the \geometrical simplicial complex") is used in the combinatorial topology. Since both terms have become standard in the appropriate science, we cannot change their names completely. We will use \polyhedron" for a geometrical simplicial complex and \convex polyhedron" for an \unbounded polytope". Anyway, it will be always clear from the context which \polyhedron" is under consideration. In the sequel both abstract and geometrical simplicial complexes are assumed to be nite. Agreement. Depending on the context, we will denote by m?1 three di erent ob jects: the abstract simplicial complex 2 m] consisting of al l subsets of m], the convex polytope from Example 1.7 (i.e. the geometrical simplex), and the geometrical simplicial complex which is the union of all faces of the geometrical simplex. Definition 2.3. Given a simplicial complex K on the vertex set m], say that a polyhedron P is a geometrical realization of K if there is a bijection between the set m] and the vertex set of P that takes simplices of K to vertex sets of faces of P . If we do not care about the dimension of the ambient space, then there is the following quite obvious way to construct a geometrical realization for any simplicial complex K . Construction 2.4. Suppose K is a simplicial complex on the set m]. Let e i denote the i-th unit coordinate vector in Rm . For each subset m] denote by the convex hull of vectors e i with i 2 . Then is a (regular, geometrical) simplex. The polyhedron is a geometrical realization of K . The above construction is just a geometrical interpretation of the fact that any simplicial complex on m] is a subcomplex of the simplex m?1 . At the same time it is a classical result 115] that any n-dimensional abstract simplicial complex K n admits a geometrical realization in (2n + 1)-dimensional space. Example 2.5. Let S be a simplicial n-polytope. Then its boundary @ S is a (geometrical) simplicial complex homeomorphic to an (n ? 1)-sphere. This example will be important in section 2.3. Definition 2.6. The f -vector, the h-vector and the g-vector of an (n ? 1)dimensional simplicial complex K n?1 are de ned in the same way as for simplicial polytopes. Namely, f (K n?1 ) = (f0 ; f1 ; : : : ; fn?1), where fi is the number of idimensional simplices of K n?1 , and h (K n?1 ) = (h0 ; h1 ; : : : ; hn ), where hi are determined by (1.7). Here we also assume f?1 = 1. If K n?1 = @ S , the boundary of a simplicial n-polytope S , then one obviously has f (K n?1 ) = f (S ).
2K

R

m


2.2. BASIC P L TOPOLOGY, AND OPERATIONS WITH SIMPLICIAL COMPLEXES

23

For a detailed exposition of P L (piecewise linear) topology we refer to the classical monographs 77] by Hudson and 118] by Rourke and Sanderson. The role of P L category in the modern topology is described, for instance, in the more recent book 104] by Novikov. Definition 2.7. Let K1, K2 be simplicial complexes on the sets m1 ], m2 ] respectively, and P1 , P2 their geometrical realizations. A map : m1 ] ! m2 ] is said to be a simplicial map between K1 and K2 if ( ) 2 K2 for any 2 K1 . A simplicial map is said to be non-degenerate if j ( )j = j j for any 2 K1 . On the geometrical level, a simplicial map extends linearly on the faces of P1 to a map : P1 ! P2 (denoted by the same letter for simplicity). We refer to the latter map as a simplicial map of polyhedrons . A simplicial isomorphism of polyhedrons is a simplicial map for which there exists a simplicial inverse. A polyhedron P 0 is called a subdivision of polyhedron P if each simplex of P 0 is contained in a simplex of P and each simplex of P is a union of nitely many simplices of P 0 . A P L map : P1 ! P2 is a map that is simplicial between some subdivisions of P1 and P2 . A P L homeomorphism is a P L map for which there exists a P L inverse. Two P L homeomorphic polyhedrons sometimes are also called combinatorial ly equivalent . In other words, two polyhedrons P1 ; P2 are P L homeomorphic if and only if there exists a polyhedron P isomorphic to a subdivision of each of them. Example 2.8. For any simplicial complex K on m] there exists a simplicial map (inclusion) K ,! m?1 . There is an obvious simplicial homeomorphism between any two geometrical realizations of a given simplicial complex K . This justi es our single notation jK j for any geometrical realization of K . Whenever it is safe, we do not distinguish between abstract simplicial complexes and their geometrical realizations. For example, we would say \simplicial complex K is P L homeomorphic to X " instead of \the geometrical realization of K is P L homeomorphic to X ". Construction 2.9 (join of simplicial complexes). Let K1 , K2 be simplicial complexes on sets S1 and S2 respectively. The join of K1 and K2 is the simplicial complex K1 K2 := S1 S2 : = 1 2 ; 1 2 K1 ; 2 2 K2 on the set S1 S2 . If K1 is realized in Rn1 and K2 in Rn2 , then there is obvious canonical geometrical realization of K1 K2 in Rn1 +n2 = Rn1 Rn2 . Example 2.10. 1. If K1 = m1 ?1 , K2 = m2 ?1 , then K1 K2 = m1 +m2 ?1 . 2. The simplicial complex 0 K (the join of K and a point) is called the cone over K and denoted cone(K ). 3. Let S 0 be a pair of disjoint points (a 0-sphere). Then S 0 K is called the suspension of K and denoted K . The geometric realization of cone(K ) (of K ) is the topological cone (suspension) over jK j. 4. Let P1 and P2 be simple polytopes. Then ? @ (P1 P2 ) = @ (P1 P2 ) = (@ P1 ) (@ P2 ): (see Construction 1.12).

2.2. Basic P L topology, and operations with simplicial complexes


24

2. TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

Construction 2.11. The fact that the product of two simplices is not a simplex causes some problems with triangulating the products of spaces. However, there is a canonical triangulation of the product of two polyhedra for each choice of orderings of their vertices. So suppose K1 , K2 are simplicial complexes on m1 ] and m2 ] respectively (this is one of the few constructions where the ordering of vertices is signi cant). Then we construct a new simplicial complex on m1 ] m2 ], which we call the Cartesian product of K1 and K2 and denote K1 K2 , as follows. By de nition, a simplex of K1 K2 is a subset of some 1 2 (with 1 2 K1, 2 2 K2 ) that establishes a non-decreasing correspondence between 1 and 2 . More formally, K1 K2 := 1 2 : 1 2 K1 ; 2 2 K2 ; and i 6 i0 implies j 6 j 0 for any two pairs (i; j ); (i0 ; j 0 ) 2 : The polyhedron jK1 K2 j de nes a canonical triangulation of jK1 j jK2 j. Construction 2.12 (connected sum of simplicial complexes). Let K1 , K2 be two pure (n ? 1)-dimensional simplicial complexes on sets S1 , S2 respectively, jS1 j = m1 , jS2 j = m2 . Suppose we are given two maximal simplices 1 2 K1 , 2 2 K2. Fix an identi cation of 1 and 2 , and denote by S1 S2 the union of S1 and S2 with 1 and 2 identi ed (the subset created by the identi cation is denoted ). We have jS1 S2 j = m1 + m2 ? n. Both K1 and K2 now can be viewed as collections of subsets of S1 S2 . We de ne the connected sum of K1 at 1 and K2 at 2 to be the simplicial complex K1 # 1 ; 2 K2 := (K1 K2) n f g on the set S1 S2 . When the choices of 1 , 2 and identi cation of 1 and 2 are clear we use the abbreviation K1 # K2 . Geometrically, the connected sum of jK1j and jK2 j at 1 and 2 is produced by attaching jK1 j to jK2 j along the faces 1 , 2 and then removing the face obtained from the identi cation of 1 with 2 . Example 2.13. 1. Let K1 be an (n ? 1)-simplex, and K2 a pure (n ? 1)dimensional complex with a xed maximal simplex 2 . Then K1 # K2 = K2 n f 2g, i.e. K1 # K2 is obtained by deleting the simplex 2 from K2 . 2. Let P1 and P2 be simple polytopes. Set K1 = @ (P1 ), K2 = @ (P2 ). Then ? K1 # K2 = @ (P1 # P2 ) (see Construction 1.13). Definition 2.14. The barycentric subdivision of an abstract simplicial complex K is the simplicial complex K 0 on the set f 2 K g of simplices of K whose simplices are chains of embedded simplices of K . That is, f 1 ; : : : ; r g 2 K 0 if and only if 1 2 r in K (after possible re-ordering). The barycenter of a (polytopal) simplex n Rn with vertices v1 ; : : : ; vn+1 1 is the point bc( n ) = n+1 (v1 + + vn+1 ) 2 n . The barycentric subdivision 0 of a polyhedron P is de ned as follows. The vertex set of P 0 is formed by the P i barycenters of simplices of P . A collection of barycenters fbc( i1 ); : : : ; bc( rr )g 1 i1 ir in P . Obviously jK 0 j = jK j0 spans a simplex of P 0 if and only if 1 r for any abstract simplicial complex K . Example 2.15. For any (n ? 1)-dimensional simplicial complex K n?1 on m] there is a non-degenerate simplicial map K 0 ! n?1 de ned on the vertices by


2.2. BASIC P L TOPOLOGY, AND OPERATIONS WITH SIMPLICIAL COMPLEXES

25

Example 2.16. Let K be a simplicial complex on a set S , and suppose we are given a choice function f : K ! S assigning to each simplex 2 K a point in . For instance, if S = m] we can take f = min, that is, assign to each simplex its minimal vertex. For every such map f there is a canonical simplicial map rf : K 0 ! K constructed as follows. By the de nition of K 0, the vertices of K 0 are in one-to-one correspondence with the simplices of K . For each 2 K (regarded as a vertex of K 0) set rf ( ) = f ( ). This extends to the simplices of K 0 by rf ( 1 2 r ) = ff ( 1 ); f ( 2 ); : : : ; f ( r )g: The latter is a subset of r , whence it is a simplex of K . Thus, rf is indeed a simplicial map. Example 2.17 (order complex of a poset). Let S be any poset. De ne ord(S ) to be the collection of all chains x1 < x2 < < xk , xi 2 S . Then ord(S ) is clearly a simplicial complex. It is called the order complex of the poset (S ; <). The order complex of the inclusion poset of non-empty simplices of a simplicial complex K is its barycentric subdivision K 0 . If we add the empty simplex to the poset, then the resulting order complex will be cone K 0 . Definition 2.18. A simplicial complex K is called a ag complex if any set of vertices which are pairwise connected spans a simplex of K . Proposition 2.19. For each simplicial graph (1-dimensional simplicial complex) ? there exists a unique ag complex K? on the same vertex set whose 1skeleton is ?. Proof. The simplices of K? are the vertex sets of complete subgraphs in ?. Definition 2.20. The minimal simplicial complex that contains a given complex K and is ag is called the agi cation of K and denoted a(K ). Definition 2.21. Given a simplicial complex K on S , a missing face of K is a subset m] such that 2 K , but every proper subset of is a simplex of K . = The following statement is straightforward. Proposition 2.22. K is a ag complex if and only if every missing face has two vertices. Example 2.23. 1. Order complexes of posets (in particular, barycentric subdivisions) are examples of ag complexes. On the other hand, the boundary of a 5-gon is ag complex, but not an order complex of poset. 2. Let K = K1 # 1 ; 2 K2 (see Construction 2.12). Then is a missing face of K provided that at least one of K1 and K2 is not a simplex. Definition 2.24. The link and the star of a simplex 2 K are the subcomplexes linkK = 2 K : 2 K; \ = ? ; starK = 2 K : 2K :

n?1 .)

! j j,

2 K . (Here is regarded as a vertex of K 0 and j j as a vertex of


26

2. TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

For any vertex v 2 K the subcomplex starK v can be identi ed with the cone over linkK v. The polyhedron j starK vj consists of all faces of jK j that contain v. We omit the subscripts K whenever the context allows. For any subcomplex L K de ne the (closed) combinatorial neighborhood UK (L) of L in K by UK (L) = starK : Equivalently, the combinatorial neighborhood UK (L) consists of all simplices of K , together with all their faces, having some simplex of L as a face. De ne also the open combinatorial neighborhood U K (L) of jLj in jK j as the union of relative interiors of faces of jK j having some simplex of jLj as their face. For any subset S de ne the ful l subcomplex K by (2.1) K = 2K : : Set core S = fv 2 S : star v 6= K g. The core of K is the subcomplex core K = Kcore S . Thus, the core is the maximal subcomplex containing all vertices whose stars do not coincide with K . Example 2.25. 1. linkK ? = K . 2. Let K = @ 3 be the boundary of the tetrahedron on four vertices 1; 2; 3; 4, and = f1; 2g. Then link is the subcomplex consisting of two disjoint points 3 and 4. 3. Let K be the cone over L with vertex v. Then link v = L, star v = K , and core K L. Example 2.26 (dual simplicial complex). Let K be a simplicial complex on S . Suppose that K is not the full simplex on S . De ne b K := S : Sn 2K : =
b Then K is also a simplicial complex on S . It is called the dual of K . b The dual simplicial complex K provides the following \purely simplicial interpretation" for the Alexander duality (see e.g. 104, p. 54]) between jK j and S m?1 n jK j for any simplicial complex K embedded in the (m ? 1)-sphere. Let us consider the barycentric subdivision (@ m?1 )0 of the boundary of a geometrical simplex on the vertex set m] = f1; : : : ; mg. By the de nition, the faces of (@ m?1 )0 correspond to the pairs of subsets of m] satisfying j j > 1, j j 6 m ? 1. Denote the corresponding faces by . (For example, fig fig is the vertex v = fig of m?1 regarded as a vertex of (@ m?1 )0 .) Denote b = m] n fig i and, more generally, b = m] n for any subset m]. For any simplicial complex K on m] de ne the following subcomplex in (@ m?1 )0 : D(K ) = b b:

2L

polyhedron D(K ) provides a geometrical realization for the barycentric subdivision of the dual simplicial complex b D(K ) = jK 0 j:

Proposition 2.27. For any simplicial complex K 6= m?1 on the set m] the

;:

; 2K =


2.2. BASIC P L TOPOLOGY, AND OPERATIONS WITH SIMPLICIAL COMPLEXES

27

Moreover, if the barycentric subdivision of K is realized canonical ly as a subpolyhedron in (@ m?1 )0 , then ? ? b jK 0 j = @ m?1 0 n U (@ m?1 )0 jK 0j : Proof. The complete proof is elementary but quite tedious. We just give an illustrating picture (Figure 2.1). Here K is the boundary of the square on vertices b 1; 2; 3; 4. Then K consists of two disjoint segments. The picture shows both K 0 0 as subcomplexes in (@ 3 )0 . b and K 2
? ? ?@ ? ?@ ?@ @ @ @ ? ? ? @ @ @ ? ? ? jK 0 j @ @ @ ? ? ? @ @ @ ? ? ? A @ @ @ ? ? ? A @ @ @ A? ? ? @ @ @ ? ? ? b 4 @ @ @ ? ? ? @ @ @ ? ? ? b jK 0 j @ @ @ ? ? ? XX X X XXX @ @ @ XX ? ? ? b XX b 3 @ @ @ X1 ? ? ? @ @ @ ? @ ? @ ? @ Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z b 2 Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z

t

1

t

d

d

4 Figure 2.1. Dual complex and Alexander duality.

t

d

d

t

3

K 6= m?1 on the set m] it holds that eb e Hj (K ) = H m?3?j (K ); ?1 6 j 6 m ? 2; e e where Hk ( ) and H k ( ) denotes the k-th reduced simplicial homology and cohomology e groups (with integer coe cients) respectively. We use the agreement H?1 (?) = e ?1 (?) = Z here. H Proof. Since (@ m?1 )0 is homeomorphic to S m?2, the Alexander duality theorem and Proposition 2.27 show that
? eb e Hj (K ) = Hj (@ Dm?1 )0 n U (@ Dm?1 )0 (jK 0 j) ? e e = Hj S m?2 n K = H m?3?j (K ); ?1 6 j 6 m ? 2:

Corollary 2.28 (Simplicial Alexander duality). For any simplicial complex

Corollary 2.28 admits the following generalization, which we will use in Chapter 8.


28

2. TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES
b 2 K it holds that

Proposition 2.29. For any simplicial complex K 6= m?1 on m] and simplex

Corollary 2.28 is obtained by substituting = ? above. b Example 2.30. Let K be the boundary of a pentagon. Then K is the Mobius band triangulated as it is shown on Figure 2.2. If we map the points b; b; b; b to the 1234 vertices of a 3-simplex and b to its barycenter, then the whole triangulated Mobius 5 b band K becomes a subcomplex in the 3-dimensional Schlegel diagram (see 145, Lecture 5]) of a 4-dimensional simplex.

? e e Hj linkK = H m?3?j?j j (Kb ); b where b = m] n and Kb is the ful l subcomplex in K de ned in (2.1).

2

r rrrr rrrrrr
3
Q Q Q
b 4 b 5 b 1

r

C C

Q Q

4

A

C C C C

A A

A

1

K

5

b 1

A A A

A A

b 2

A A A

b 3

A A U A A A A

b 4

b K

Figure 2.2. The boundary of pentagon and its dual.

Definition 2.31. A simplicial q-sphere is a simplicial complex K q homeomorphic to q-sphere. A P L sphere is a simplicial sphere K q which is P L homeomorphic to the boundary of a simplex (equivalently, there is a subdivision of K q isomorphic to a subdivision of the boundary of q+1 ). A homology q-sphere is a topological manifold which has the same homology as the q-sphere S q .

2.3. Simplicial spheres

The boundary of a simplicial n-polytope is an (n ? 1)-dimensional P L sphere. A P L sphere simplicially isomorphic to the boundary of a simplicial polytope is called a polytopal sphere . We have the following hierarchy of combinatorial ob jects: (2.2) polytopal spheres P L spheres simplicial spheres. In dimension 2 any simplicial sphere is polytopal (see e.g. 69] or 145, Theorem 5.8]). However, in higher dimensions both above inclusions are strict. The rst inclusion in (2.2) is strict already in dimension 3. Namely, there are 39 combinatorially di erent triangulations of the 3-sphere with 8 vertices, out of which 2 are non-polytopal . The rst one, now known as the Bruckner sphere was found by Grunbaum ( 69, x11.5], see also 70]) as a correction of Bruckner's result of 1909 on the classi cation of simplicial 4-polytopes with 8 vertices. The second, known as Barnette sphere , is described in 12]. The complete classi cation of simplicial 3-spheres with up to 8 vertices was obtained in 14]. We mention also the result of Mani 94] that any simplicial q-sphere with up to (q + 4) vertices is polytopal.


2.4. TRIANGULATED MANIFOLDS

29

As for the second inclusion in (2.2), it is known that in dimension 3 any simplicial sphere is P L. In dimension 4 the corresponding question is open (see the discussion in the next section), but starting from dimension 5 there exist non-P L simplicial spheres. One such thing is described in Example 2.35 below. According to the result of 21], for any n > 5 there is a non-P L triangulation of S n with n + 13 vertices. Since the f -vector of a polytopal sphere coincides with the f -vector of the corresponding simplicial polytope (see De nition 2.6), the g-theorem (Theorem 1.29) holds for polytopal spheres. So it is natural to ask whether the g-theorem extends to simplicial spheres. This question was posed by McMullen 96] as an extension of his conjecture for simplicial polytopes. Since 1980, when McMullen's conjecture for simplicial polytopes was proved by Billera, Lee, and Stanley, the following is regarded as the main open combinatorial-geometrical problem concerning the f -vectors of simplicial complexes. Problem 2.32 (g-conjecture for simplicial spheres). Does the g-theorem (Theorem 1.29) hold for simplicial spheres? The g-conjecture is open even for P L spheres. Note that only the necessity of g-theorem (i.e. that the g-vector is an M -vector) is to be veri ed for simplicial spheres. If correct, the g-conjecture would imply a characterisation of f -vectors of simplicial spheres. The rst part of Theorem 1.29 (the Dehn{Sommerville equations) is known to be true for simplicial spheres (see Corollary 3.41 below). Simplicial spheres also satisfy the UBC and the LBC inequalities as stated in Theorems 1.33 and 1.37. The LBC (in particular, the inequality h1 6 h2 ) for spheres was proved by Barnette 15] (see also 83]). The UBC for spheres is due to Stanley 123] (see Corollary 3.19 below). This implies that the g-conjecture is true for simplicial spheres of dimension 6 4. The inequality h2 6 h3 from the GLBC (1.14) is open. Many attempts to prove the g-conjecture were made during the last two decades. Though unsuccessful, these attempts resulted in some very interesting reformulations of the g-conjecture. The results of Pachner 109], 110] reduce the g-conjecture (for P L-spheres) to some properties of bistel lar moves (see the discussion after Theorem 2.41). We also mention the results of 131] showing that the g-conjecture follows from the skeletal r-rigidity of simplicial (n ? 1)-sphere for r 6 n . It was shown independently by 2 Kalai and Stanley 127, Corollary 2.4] that the GLBC holds for the boundary of an n-dimensional ball that is a subcomplex of the boundary complex of a simplicial (n + 1)-polytope. However, it is not clear now which simplicial complexes occur in this way. The lack of progress in proving the g-conjecture motivated Bjorner and Lutz to launch a computer-aided search for counterexamples 21]. Though their bistellar ip algorithm and computer program BISTELLAR produced many remarkable results on triangulations of manifolds, no counterexamples to the g-conjecture were found. For more history of g-theorem and related questions see 128], 129], 145, Lecture 8].
Definition 2.33. A simplicial complex K is called a triangulated manifold (or simplicial manifold ) if the polyhedron jK j is a topological manifold. (All manifolds considered here are compact, connected and closed, unless otherwise stated.) A q-dimensional P L manifold (or combinatorial manifold ) is a simplicial complex K q

2.4. Triangulated manifolds


30

2. TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

such that link( ) is a P L sphere of dimension (q ? j j) for every non-empty simplex 2 Kq. Every P L manifold K q is a (triangulated) manifold. Indeed, for each vertex v 2 K q the (q ? 1)-dimensional P L-sphere link v bounds an open neighborhood Uv which is homeomorphic to an open q-ball. Since any point of jK q j is contained in Uv for some v, this de nes an atlas for jK q j. Does every triangulation of a topological manifold yield a simplicial complex which is a P L manifold? The answer is \no", and the question itself ascends to a famous conjecture of the dawn of topology, known as the Hauptvermutung der Topologie . In the early days of topology all of the known topological invariants were de ned in combinatorial terms, and it was very important to nd out whether the topology of a polyhedron fully determines the combinatorics of triangulation. In the modern terminology, the Hauptvermutung conjecture states that any two homeomorphic polyhedrons are combinatorially equivalent (P L homeomorphic). This is valid in dimensions 6 3 (the result is due to Rado, 1926, for 2-manifolds, Papakyriakopoulos, 1943, for 2-complexes, Moise, 1953 for 3-manifolds, and E. Brown, 1964, for 3-complexes; see 101] for the modern exposition). The rst examples of complexes disproving the Hauptvermutung in dimensions > 6 were found by Milnor in the early 1960s. However, the manifold Hauptvermutung , namely the question of whether two homeomorphic triangulated manifolds are combinatorially equivalent, had remained open until the 1970s. It was nally disproved with the appearance of the following theorem. n Theorem 2.34 (Edwards, Cannon). The double suspension SH of any hon is homeomorphic to S n+2 . mology n-sphere SH This theorem was proved by Edwards 58] for some particular homology 3spheres and by Cannon 39] in the general case. The following example provides a non-P L triangulation of the 5-sphere and therefore disproves the manifold Hauptvermutung in dimensions > 5. 3 Example 2.35 (non-P L simplicial 5-sphere). Let SH be any simplicial homology 3-sphere which is not a topological sphere. The Poincare sphere S O(3)=A5 (triangulated in any way) provides an example of such a manifold. By Theorem 2.34, 3 3 the double suspension 2 SH is homeomorphic to S 5 (and, more generally, k SH k+3 for k > 2). However, 2 S 3 cannot be P L, since S 3 is homeomorphic to S H H 3 appears as the link of some 1-simplex in 2 SH . In the positive direction, it is known that two homeomorphic simply connected P L manifolds of dimension > 5 with no torsion in third homology group are combinatorially equivalent (P L homeomorphic). This is Sullivan's famous Hauptvermutung theorem. The general classi cation of P L structures on higher dimensional topological manifolds was obtained by Kirby and Siebenmann, see 85]. The following theorem gives a characterization of simplicial complexes which are triangulated manifolds of dimension > 5 and generalizes Theorem 2.34. Theorem 2.36 (Edwards 59]). For q > 5 the polyhedron of a simplicial complex K q is a topological q-manifold if and only if link has the homology of a (q ? j j)-sphere for each non-empty simplex j j 2 K q and link v is simply connected for each vertex v 2 K .


2.5. BISTELLAR MOVES

31

The discovery of non-P L triangulations of topological manifolds motivated further questions. Among them is whether every topological manifold admits a P L triangulation, or at least any triangulation, not necessarily P L. Another related question is whether the Hauptvermutung is valid in dimension 4. Both questions were answered (negatively) by the results of Freedman and Donaldson (early 1980s). A smooth manifold can be triangulated by Whitney's theorem. All topological 2- and 3-dimensional manifolds can be triangulated as well (for 3-manifolds see 101]). Moreover, since the link of a vertex in a simplicial 3-sphere is a 2-sphere (and a 2-sphere is always P L), all 2- and 3-manifolds are P L. However, in dimension 4 there exist topological manifolds that do not admit a P L-triangulation. An example is provided by Freedman's fake C P 2 63, x8.3, x10.1], a topological manifold which is homeomorphic, but not di eomorphic to the complex pro jective plane C P 2 . This shows that the Hauptvermutung is false for 4-dimensional manifolds. Even worse, as it is shown in 5], there exist topological 4-manifolds (e.g. Freedman's topological 4-manifold with the intersection form E8 ) that do not admit any triangulation . In dimensions > 5 the triangulation problem is open: Problem 2.37 (Triangulation Conjecture). Is it true that any topological manifold of dimension > 5 can be triangulated? Another well-known problem of P L-topology concerns the uniqueness of a P L structure on a topological sphere. Problem 2.38. Is a P L manifold homeomorphic to the topological 4-sphere necessarily a P L sphere? Four is the only dimension where the uniqueness of a P L structure on a topological sphere is open. For dimensions 6 3 the uniqueness was proved by Moise 101], and for dimensions > 5 it follows from the result of Kirby and Siebenmann 85]. In dimension 4 the category of P L manifolds is equivalent to the smooth category, hence, the above problem is equivalent to if there exists an exotic (or fake) 4-sphere. The history of the Hauptvermutung conjecture is summarized in a survey article 116] by A. Ranicki. This source also contains a more detailed discussion of recent developments and open problems (including those mentioned above) in combinatorial and P L topology. Bistellar moves (in other notation, bistellar ips or bistellar operations) were introduced by Pachner (see 109], 110]) as a generalization of stel lar subdivisions . These operations allow us to decompose a P L homeomorphism into a sequence of simple \ ips" and thus provide a very convenient way to compute and handle topological invariants of P L manifolds. Starting from a given P L triangulation, bistellar operations may be used to construct new triangulations with some good properties, e.g. symmetric or with a small number of vertices. On the other hand, bistellar moves can be used to produce some nasty triangulation if we start from a non-P L one. Both approaches were applied in 21] to nd many interesting triangulations of low-dimensional manifolds. Bistellar moves also provide a combinatorial interpretation for algebraic blow up and blow down operations for pro jective toric varieties (see section 5.1) as well as for some topological surgery operations (see Construction 6.23). Finally, bistellar moves may be used to de ne a metric on the space of P L triangulations of a given P L manifold, see 103] for more details.

2.5. Bistellar moves


32

2. TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES

Definition 2.39. Let K be a simplicial q-manifold (or any pure q-dimensional simplicial complex) and 2 K a (q ? i)-simplex (0 6 i 6 q) such that linkK is the boundary @ of an i-simplex that is not a face of K . Then the operation on K de ned by ? (K ) := K n ( @ ) (@ ) q are also called reverse is called a bistel lar i-move. Bistellar i-moves with i > 2 (q ? i)-moves . Note that a 0-move adds a new vertex to a triangulation (we assume that @ D0 = ?), a reverse 0-move deletes a vertex, while all other bistellar moves do not change the number of vertices, see Figures 2.3 and 2.4. Two pure simplicial complexes are bistel larly equivalent if one is taken to another by a nite sequence of bistellar moves.
S S S

0-move 2-move
-

S

S

S PP S PP P S ?@ @

S

S

S S

? ? @ @

?

?

@

@

@

? @ @?

?

@ ? ?

1-move

-

? ? @ @

? ?

?@ @

@ @ ?

@ ? ?

@

@ @?

?

Figure 2.3. Bistellar moves for q = 2. Remark. The bistellar 0-move is just the stellar subdivision, or connected sum with the boundary of a simplex. In particular, stacked spheres (i.e., the boundaries of stacked polytopes, see De nition 1.36) are exactly those obtained from the boundary of a simplex by applying bistellar 0-moves. It is easy to see that two bistellarly equivalent P L manifolds are P L homeomorphic. The following remarkable result shows that the converse is also true. Theorem 2.40 (Pachner 109, Theorem 1], 110, (5.5)]). Two P L manifolds are bistel larly equivalent if and only if they are P L homeomorphic. The behavior of the face numbers of a triangulation under bistellar moves is easily controlled. Namely, the following statement holds. Theorem 2.41 (Pachner 109]). Let L be a q-dimensional triangulated manifold obtained from K by applying a bistel lar k-move, 0 6 k 6 q?1 . Then 2 gk+1 (L) = gk+1 (K ) + 1; gi (l) = gi (K ) for al l i 6= k + 1; where gi (K ) = hi (K ) ? hi?1 (K ), 0 < i 6 n , are the components of the g-vector. 2 q Moreover, if q is even and k = 2 , then gi (L) = gi (K ) for al l i.


2.5. BISTELLAR MOVES
J @ J @ J J@ @ JH @ H@ H Q Q Q ?@ @

33
@ @

0-move 3-move
Q Q Q

@

@

Q Q

@ @

Q

Q ?@ @

? ? @ H H ? @H ? @ HH ? H @ ? B @ ? @B ? @B ? B @ @? B

?

?

?

?

?

@

@

@

@ @

1-move 2-move
-

@ ? ? @ H H @H ? ? @ HH ? H @ ? B @ ? @B ? @B ? B @ @? B

?

?

? ?

?

@

@

@

@

Figure 2.4. Bistellar moves for q = 3.

with boundary , see 110, (6.3)]. For this purpose he introduced another class of operations on triangulations, called elementary shel lings .

This theorem allows us to interpret the inequalities from the g-conjecture for P L spheres (see Theorem 1.29) in terms of the numbers of bistellar k-moves needed to transform the given P L sphere to the boundary of a simplex. For instance, the inequality h1 6 h2 , n > 4, is equivalent to the statement that the number of 1-moves in the sequence of bistellar moves taking a given (n ? 1)-dimensional P L sphere to the boundary of an n-simplex is lesser than or equal to the number of reverse 1-moves. (Note that the g-vector of @ n has the form (1; 0; : : : ; 0).) Remark. Pachner also proved an analogue of Theorem 2.40 for P L manifolds


34

2. TOPOLOGY AND COMBINATORICS OF SIMPLICIAL COMPLEXES


CHAPTER 3

Commutative and homological algebra of simplicial complexes
The appearance of the Stanley{Reisner face ring of simplicial complex at the beginning of 1970s outlined a new approach to combinatorial problems concerning simplicial complexes. It relies upon the interpretation of combinatorial properties of simplicial complexes as algebraic properties of the corresponding face rings and uses commutative algebra machinery such as Cohen{Macaulay and Gorenstein algebras, local cohomology, etc. The main reference here is R. Stanley's monograph 128].

Definition 3.1. The face ring (or the Stanley{Reisner ring ) of a simplicial complex K on the vertex set m] is the quotient ring k(K ) = k v1 ; : : : ; vm]=IK ; where IK is the homogeneous ideal generated by all square-free monomials v = vi1 vi2 vis (i1 < < is ) such that = fi1 ; : : : ; is g is not a simplex of K . The ideal IK is called the Stanley{Reisner ideal of K . Suppose P is a simple n-polytope, P its polar, and KP the boundary of P . Then KP is a polytopal simplicial (n ? 1)-sphere. The face ring of P from De nition 1.38 coincides with that of KP from the above de nition: k(P ) = k(KP ). Example 3.2. 1. Let K be a 2-dimensional simplicial complex shown on Figure 3.1. Then IK = (v1 v5 ; v3 v4 ; v1 v2 v3 ; v2 v4 v5 ):

3.1. Stanley{Reisner face rings of simplicial complexes Recall that k v1 ; : : : ; vm ] denotes the graded polynomial algebra over a commutative ring k with unit, deg vi = 2.

1

?B ?@ B ? B@ ? B@ ? BH @ @ ? 4 5 HH @ ? HH @ ? H @

r

rr
35

2

r

r

3

Figure 3.1


36

3. COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

2. The Stanley{Reisner ring k(K ) is a quadratic algebra (i.e. the ideal IK is generated by quadratic monomials) if and only if K is a ag complex (see De nition 2.18 and compare with Proposition 2.19). 3. Let K1 K2 be the join of K1 and K2 (see Construction 2.9). Then k(K1 K2) = k(K1) k(K2): In particular, for any two simple polytopes P1 and P2 we have k(P1 P2 ) = k(P1 ) k(P2 ) (see Construction 1.12). 4. Let K1 # 1 ; 2 K2 be the connected sum of two pure (n ? 1)-dimensional simplicial complexes on sets S1 , S2 respectively, regarded as a simplicial complex on the set S1 S2 (see Construction 2.12). Then the ideal IK1 # 1 ; 2 K2 is generated by the ideals IK1 , IK2 and the monomials v and vi1 vi2 , where i1 2 S1 n 1 and i2 2 S2 n 2 . For any subset = fi1; : : : ; ik g m] denote by v the square-free monomial vi1 vik . Note that the ideal IK is monomial and has basis of monomials v corresponding to missing faces of K . Proposition 3.3. Every square-free monomial ideal in the polynomial ring has the form IK for some simplicial complex K . Proof. Let I be a square-free monomial ideal. Set K=f m] : v 2 I g: = Then one easily checks that K is a simplicial complex and I = IK .
Proposition 3.4. Let : K1 ! K2 be a simplicial map (see De nition 2.7) between two simplicial complexes K1 and K2 on the vertex sets m1 ] and m2 ] respectively. De ne the map : k w1 ; : : : ; wm2 ] ! k v1 ; : : : ; vm1 ] by

(wj ) :=

X

Then descends to a homomorphism k(K2 ) ! k(K1 ) (which we wil l also denote by ). Proof. We have to check that (IK2 ) IK1 . Suppose = fj1 ; : : : ; js g m2 ] is not a simplex of K2 . Then

fig2f ?1 fjg

vi :

(3.1)

(wj

1

wjs ) =

X

We claim that = fi1; : : : ; is g is not a simplex of K1 for any monomial vi1 vis in the right hand side of the above identity. Indeed, otherwise we would have ( ) = 2 K2 by the de nition of simplicial map, which is impossible. Hence, the right hand side of (3.1) is in IK1 .
Example 3.5. The face ring of the barycentric subdivision K 0 of K is k(K 0) = k b : 2 K ]=IK0 ;

fi1 g2 ?1 fj1 g;::: ;fis g2 ?1 fjs g

vi1

vis :


3.1. STANLEY{REISNER FACE RINGS OF SIMPLICIAL COMPLEXES

37

where b is the polynomial generator corresponding to simplex 2 K . We have the simplicial map r : K 0 ! K (see Example 2.16). Then it is easy to see that

r (vj ) :=

X

for any generator vj 2 k(K ). Example 3.6. The non-degenerate map duces the following map of the corresponding k v1 ; : : : ; vn] ?!

2K : min =j

b:

vi ?!

Stanley{Reisner rings: k(K 0)
X

K 0 ! n?1 from Example 2.15 inb:

This de nes a canonical k v1 ; : : : ; vn ]-module structure in k(K 0 ). Definition 3.7. Let M = M 0 M 1 : : : be a graded k-module. The series

j j=i

F (M ; t) =

1 X
i=0

(dimk M i )ti

where (f0 ; : : : ; fn?1) is the f -vector and (h0 ; : : : ; hn ) is the h-vector of K n?1. Proof. Any monomial in k(K n?1 ) has the form vi11 vikk+1 , where +1 fi1; : : : ; ik+1 g is a simplex of K n?1 and 1 ; : : : ; k+1 are some positive integers. 2( Thus, every k-simplex of K n?1 contributes the summand (1t?tk2+1)+1 to the Poincare )k series, which proves the rst identity. The second identity is an obvious corollary of (1.8).
? +1 Example 3.9. 1. Let K = n . Then fi = n+1 for ?1 6 i 6 i and hi = 0 for i > 0. Since any subset of n + 1] is a simplex of k( n ) = k v1 ; : : : ; vn+1] and F (k( n ); t) = (1 ? t2)?(n+1), which

is called the Poincare series of M . Remark. In the algebraic literature the series F (M ; t) is called the Hilbert series or Hilbert{Poincare series . The following lemma may be considered as an algebraic de nition of the hvector of a simplicial complex. Lemma 3.8 (Stanley 128, Theorem II.1.4]). The Poincare series of k(K n?1 ) can be calculated as n?1 f t2(i+1) 2n X ? t2 + i F k(K n?1 ); t = = h0 + h1(1 ? t2 )n+ hn t ; 2 i+1 i=?1 (1 ? t )

agrees with Lemma 3.8. 2. Let K be the boundary of an n-simplex. Then hi = 1, i = 0; 1; : : : ; n, and k(K ) = k v1 ; : : : ; vn+1]=(v1 v2 vn+1). By Lemma 3.8, 2 ? + 2n F k(K ); t = 1 + t(1 + t2 )n t : ?

n , we have

n, h0 = 1


38

3. COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

so it is commutative in both the usual and the graded sense. Definition 3.10. The Krul l dimension of A (denoted Kd A) is the maximal number of algebraically independent elements of A. A sequence 1 ; : : : ; n of n = Kd A homogeneous elements of A is called an hsop (homogeneous system of parameters) if the Krull dimension of the quotient A=( 1 ; : : : ; n ) is zero. Equivalently, 1 ; : : : ; n is an hsop if n = Kd A and A is a nitely-generated k 1 ; : : : ; n ]module. The elements of an hsop are algebraically independent. Lemma 3.11 (Noether normalization lemma). For any nitely-generated graded algebra A there exists an hsop. If k is of zero characteristic and A is generated by degree-two elements, then a degree-two hsop can be chosen. In the case when A is generated by degree-two elements, a degree-two hsop is called an lsop (linear system of parameters). Remark. If k is of nite characteristic then an lsop may fail to exist for algebras generated in degree two, see Example 5.26 below. In the rest of this chapter we assume that the eld k is of zero characteristic. Definition 3.12. A sequence 1 ; : : : ; k of homogeneous elements of A is called a regular sequence if i+1 is not a zero divisor in A=( 1 ; : : : ; i ) for 0 6 i < k (i.e. the multiplication by i+1 is a monomorphism of A=( 1 ; : : : ; i ) into itself ). Equivalently, 1 ; : : : ; k is a regular sequence if 1 ; : : : ; k are algebraically independent and A is a free k 1 ; : : : ; k ]-module. Remark. The concept of a regular sequence can be extended to non- nitelygenerated graded algebras and to algebras over any integral domain. Regular sequences in graded polynomial rings R a1 ; a2 ; : : : ; ] on in nitely many generators, where deg ai = ?2i and R is a subring of the eld Q of rationals, are used in the algebraic topology for constructing complex cobordism theories with coe cients, see 89]. Any two maximal regular sequences have the same length, which is called the depth of A and denoted depth A. Obviously, depth A 6 Kd A. Definition 3.13. Algebra A is called Cohen{Macaulay if it admits a regular sequence 1 ; : : : ; n of length n = Kd A. A regular sequence 1 ; : : : ; n of length n = Kd A is an hsop. It follows that A is Cohen{Macaulay if and only if there exists an hsop 1 ; : : : ; n such that A is a free k 1 ; : : : ; n ]-module. If in addition A is generated by degree-two elements, then one can choose 1 ; : : : ; n to be an lsop. In this case the following formula for the Poincare series of A holds ? F A=( 1 ; : : : ; n ); t F (A; t) = ; (1 ? t2 )n where F (A=( 1 ; : : : ; n ); t) = h0 + h1 t2 + is a polynomial. The nite vector (h0 ; h1 ; : : : ) is called the h-vector of A.

3.2. Cohen{Macaulay rings and complexes Here we suppose k is a eld. Let A be a nitely-generated commutative graded algebra over k. We also assume that A has only even-degree graded components,


3.2. COHEN{MACAULAY RINGS AND COMPLEXES

39

k

coincides with the h-vector of K . Example 3.15. Let K 1 be the boundary of a 2-simplex. Then k(K 1 ) = k v1 ; v2; v3]=(v1 v2v3 ). The elements v1; v2 2 k(K ) are algebraically independent, but do not form an hsop, since k(K )=(v1 ; v2 ) = k v3 ] and Kd k(K )=(v1 ; v2 ) = 1 = 0. 6 On the other hand, the elements 1 = v1 ? v3 , 2 = v2 ? v3 of k(K ) form an hsop, since k(K )=( 1 ; 2 ) = k t]=t3 . It is easy to see that k(K ) is a free k 1 ; 2 ]-module with one 0-dimensional generator 1, one 1-dimensional generator v1 , and one 22 dimensional generator v1 . Thus, k(K ) is Cohen{Macaulay and ( 1 ; 2 ) is a regular sequence. Theorem 3.16 (Stanley). If K n?1 is a Cohen{Macaulay simplicial complex, then h(K n?1 ) = (h0 ; : : : ; hn ) is an M -vector (see De nition 1.31). Proof. Let 1 ; : : : ; n be a regular sequence of degree-two elements of k(K ). Then A = k(K )=( 1 ; : : : ; n ) is a graded algebra generated by degree-two elements, and dimk A2i = hi . Now the result follows from Theorem 1.32. The following fundamental theorem characterizes Cohen{Macaulay complexes combinatorially. Theorem 3.17 (Reisner 117]). A simplicial complex K is Cohen{Macaulay over k if and only if for any simplex 2 K (including = ?) and i < dim(link ), e e Hi (link ; k) = 0. (Here Hi (X ; k) denotes the i-th reduced homology group of X with coe cients in k.) Corollary 3.18. A simplicial sphere is a Cohen{Macaulay complex. Theorem 3.16 shows that the h-vector of a simplicial sphere is an M -vector. This argument was used by Stanley to extend the UBC (Theorem 1.33) to simplicial spheres. Corollary 3.19 (Upper Bound Theorem for spheres, Stanley 123]). The hvector (h0 ; h1 ; : : : ; hn ) of a simplicial (n ? 1)-sphere K n?1 with m vertices satis es
? hi (K n?1 ) 6 m?ni+i?1 ;

Definition 3.14. A simplicial complex K n?1 is called Cohen{Macaulay (over ) if its face ring k(K n?1 ) is Cohen{Macaulay. Obviously, Kd k(K n?1 ) = n. Lemma 3.8 shows that the h-vector of k(K )

06i< n : 2

Hence, the UBC holds for simplicial spheres, that is,
? fi (K n?1 ) 6 fi C n (m)

for i = 2; : : : ; n ? 1:

(see Corol lary 1.35).

(Theorem 1.32). In particular, dimk A2 = h1 = m ? n. Since A is generated by A2 , the number hi cannot exceed the total number of monomials of degree i in (m ? n) ? variables. The latter is exactly m?ni+i?1 .

Proof. Since h (K n?1 ) is an M -vector, there exists a graded algebra A = A A2 A2n generated by degree-two elements such that dimk A2i = hi
0


40

3. COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

Here we review some homological algebra. Unless otherwise stated, all modules in this section are assumed to be nitely-generated graded k v1 ; : : : ; vm ]-modules, deg vi = 2. Definition 3.20. A nite free resolution of a module M is an exact sequence (3.2) 0 ! R?h ?d R?h+1 ?d ! ! ?! R?1 ?d R0 ?d M ! 0; ! ! ?i are nitely-generated free modules and the maps d are degreewhere the R preserving. The minimal number h for which a free resolution (3.2) exists is called the homological dimension of M and denoted hd M . By the Hilbert syzygy theorem a nite free resolution (3.2) exists and hd M 6 m. A resolution (3.2) can be written L as a free bigraded di erential module R; d], where R = R?i;j , R?i;j := (R?i )j (the j -th graded component of the free module R?i ). The cohomology of R; d] is zero in non-zero dimensions and H 0 R; d] = M . Conversely, a free bigraded L di erential module R = i;j>0 R?i;j ; d : R?i;j ! R?i+1;j ] with H 0 R; d] = M ?i := R?i; = and ?i L H i;j R; d] = 0 for i > 0 de nes a free resolution (3.2) with R ?. jR Remark. For the reasons speci ed below we numerate the terms of a free resolution by non-positive numbers, thereby viewing it as a cochain complex. The Poincare series of M can be calculated from any free resolution (3.2) by means of the following classical theorem. Theorem 3.21. Suppose that R?i has rank qi with free generators in degrees d1i ; : : : ; dqi i , i = 1; : : : ; h. Then (3.3)

3.3. Homological algebra background

F (M ; t) = (1 ? t2 )?m

h X i=0

(?1)i (td1i +

+ tdqi i ):

Proof. By the de nition of resolution, the following map of cochain complexes 0 ? ?? R?h ? ?? R?h+1 ? ?? ?! ?d ! ? d ! ? ?? R?1 ? ?? R0 ? ?? 0 ?d ! ?d ! ?!
? ? y ? ? y ? ? y ? ? y

0 ? ?? 0 ? ?? ?! ? ! 0 ? ?? ? ! ? ?? 0 ? ?? M ? ?? 0 ?! ?! ?! is a quasi-isomorphism, i.e. induces an isomorphism in the cohomology. Equating the Euler characteristics of both complexes in each degree we get (3.3) Construction 3.22. There is the following straightforward way to construct a free resolution for a module M . Take a set of generators a1 ; : : : ; ak0 for M and de ne R0 to be a free k v1 ; : : : ; vm ]-module with k0 generators in the corresponding degrees. There is an obvious epimorphism R0 ! M . Then take a set of generators a1 ; : : : ; ak1 in the kernel of R0 ! M and de ne R?1 to be a free k v1 ; : : : ; vm ]module with k1 generators in the corresponding degrees, and so on. On the i-th step we take a set of generators in the kernel of the previously constructed map d : R?i+1 ! R?i+2 and de ne R?i to be a free module with the corresponding generators. The Hilbert syzygy theorem guarantees this process to end up at most at the m-th step.


3.3. HOMOLOGICAL ALGEBRA BACKGROUND

41

M a minimal generator set (or a minimal basis ) can be chosen. This is done as follows. Take the lowest degree in which M is non-zero and there choose a vector space basis. Span a module M1 by this basis and then take the lowest degree in which M 6= M1 . In this degree choose a vector space basis in the complement of M1 , and span a module M2 by this basis and M1 . Then continue this process. Since M is nitely generated, on some p-th step we get M = Mp and a basis for M
with a minimal number of generators. If we take a minimal set of generators for modules at each step of Construction 3.22, then the produced resolution is called minimal . Each of its terms R?i has the smallest possible rank (see Example 3.26 below). There is also the following more formal (but less convenient for particular computations) de nition of minimal resolution (see 2]). Let M , M 0 be two modules. Set J (M ) = v1 M + v2 M + + vm M M . A map f : M ! M 0 is called minimal if Ker f J (M ). A resolution (3.2) is called minimal if all maps d are minimal. A minimal resolution is unique up to an isomorphism. Example 3.24 (Koszul resolution). Let M = k. The k v1 ; : : : ; vm ]-module structure on k is de ned via the map k v1 ; : : : ; vm ] ! k that sends each vi to 0. Let u1; : : : ; um] denote the exterior algebra on m generators. Turn the tensor product R = u1 ; : : : ; um ] k v1 ; : : : ; vm ] (here and below we use for k) into a di erential bigraded algebra by setting bideg ui = (?1; 2); bideg vi = (0; 2); (3.4) dui = vi ; dvi = 0; and requiring that d be a derivation of algebras. An explicit construction of cochain homotopy 92, x7.2] shows that H ?i R; d] = 0 for i > 0 and H 0 R; d] = k. Since u1; : : : ; um] k v1 ; : : : ; vm ] is a free k v1 ; : : : ; vm ]-module, it determines a free resolution of k. This resolution is known as the Koszul resolution . Its expanded form is as follows:

Example 3.23 (minimal resolution). For graded nitely generated modules

0 ! m u1 ; : : : ; um ] k v1 ; : : : ; vm ] ?! ?! 1 u1 ; : : : ; um ] k v1 ; : : : ; vm ] ?! k v1 ; : : : ; vm ] ?! k ! 0; where i u1 ; : : : ; um] is the submodule of u1; : : : ; um] spanned by monomials of length i. Thus, in the notation of (3.2) we have R?i = i u1 ; : : : ; um ] k v1 ; : : : ; vm]. Let N be another module; then applying the functor k v1 ;::: ;vm ] N to (3.2) we obtain the following cochain complex of graded modules: 0 ?! R?h k v1 ;::: ;vm ] N ?! ?! R0 k v1;::: ;vm ] N ?! 0 and the corresponding bigraded di erential module R N ; d]. The (?i)-th cohomology module of the above cochain complex is denoted Tor?iv1 ;::: ;vm ] (M ; N ), k i.e. Tor?iv1 ;::: ;vm ] (M ; N ) := H ?i R k v1 ;::: ;vm ] N ; d] k Ker d : R?i k v1 ;::: ;vm ] N ! R?i+1 k v1 ;::: ;vm ] N ] = : d(R?i?1 k v1;::: ;vm ] N )


42

3. COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

Since both the R?i 's and N are graded modules, we actually have M ? Tork iv1 ;::: ;vm ] (M ; N ) = Tor?i;j;::: ;vm ] (M ; N ); k v1 where
j ;::: ;vm ] N )j ! (R?i+1 k v1 ;::: ;vm ] N )j : d(R?i?1 k v1;::: ;vm ] N )j The above modules combine to a bigraded k v1 ; : : : ; vm ]-module, M Tork v1 ;::: ;vm ] (M ; N ) = Tor?i;j;::: ;vm ] (M ; N ): k v1 i;j The following properties of Tor?iv1 ;::: ;vm ] (M ; N ) are well known (see e.g. 92]). k Proposition 3.25. (a) The module Tor?iv1 ;::: ;vm ] (M ; N ) does not depend, up k
k v1

Ker d : (R?i Tor?i;j;::: ;vm ] (M ; N ) = k v1

to isomorphism, on a choice ? (b) Both Tork iv1 ;::: ;vm ] ( (c) Tor0 v1 ;::: ;vm ] (M ; N ) k (d) Tor?iv1 ;::: ;vm ] (M ; N ) k

In homological algebra A-modules TorA (M ; N ) are de ned for algebras A far more general than polynomial rings (and nitely-generatedness assumption for modules M and N may be also dropped). Although a nite A-free resolution (3.2) of M may fail to exist in general, there is always a projective resolution, which allows one to de ne TorA (M ; N ) in the same way as above. Note that pro jective modules over the polynomial algebra are free. In the non-graded case this was known as the Serre problem , now solved by Quillen and Suslin. However the graded version of this fact is much easier to prove. In this text the Tor-modules TorA (M ; N ) over algebras di erent from the polynomial ring appear only in sections 7.1 and 8.3.

of resolution (3.2). ; N ) and Tor?iv1 ;::: ;vm ] (M ; ) are covariant functors. k = M k v1;::: ;vm ] N . = Tor?iv1 ;::: ;vm ] (N ; M ). k

3.4. Homological properties of face rings: Tor-algebras and Betti numbers
m M

M = k(K ) and N = k. As usual, K = K n?1 is assumed to be a simplicial complex on m]. Since deg vi = 2, we have
Tork v1 ;::: ;v
m]
?

Here we apply general constructions from the previous section in the case when

k(K ); k =

(i.e. Tork v1 ;::: ;vm ] (k(K ); k) is non-zero only in even second degrees). De ne the bigraded Betti numbers of k(K ) by ? ?i;2j ?k(K ) := dimk Tor?i;2j (3.5) 0 6 i; j 6 m: k v1 ;::: ;vm ] k(K ); k ; Suppose that (3.2) is a minimal free resolution of M = k(K ) (Example 3.23). Then R0 = k v1 ; : : : ; vm ] is a free k v1 ; : : : ; vm ]-module with one generator of degree 0. The basis of R?1 is a minimal generator set for IK = Ker k v1 ; : : : ; vm ] ! k(K )] and is represented by the missing faces of K . For each missing face fi1; : : : ; ik g of K denote by vi1 ;::: ;ik the corresponding basis element of R?1 . Then deg vi1 ;::: ;ik = 2k

i;j =0

2j Tor?i;1 ;::: ;v kv

m]

?

k(K ); k


3.4. HOMOLOGICAL PROPERTIES OF FACE RINGS

43

and the map d : R?1 ! R0 takes vi1 ;::: ;ik to vi1 vik . Since the maps d in (3.2) are minimal, the di erentials in the cochain complex 0 ?! R?h k v1;::: ;vm ] k ?! ?! R0 k v1;::: ;vm ] k ?! 0 are trivial. Hence, for the minimal resolution of k(K ) it holds that ? (3.6) Tor?iv1 ;::: ;vm ] k(K ); k = R?i k v1 ;::: ;vm ] k; k ?i;2j ?k(K ) = rank R?i;2j : Example 3.26. Let K 1 be the boundary of a square. Then k(K 1 ) = k v1 ; : : : ; v4 ]=(v1v3; v2 v4): Let us construct a minimal resolution of k(K 1 ) using Construction 3.22. The module R0 has one generator 1 (of degree 0), and the map R0 ! k(K 1 ) is the quotient pro jection. Its kernel is the ideal IK 1 , and the minimal basis consists of two monomials v1 v3 and v2 v4 . Hence, R?1 has two free generators of degree 4, denoted v13 and v24 , and the map d : R?1 ! R0 sends v13 to v1 v3 and v24 to v2 v4 . The minimal basis for the kernel of R?1 ! R0 consists of one element v2 v4 v13 ? v1 v3 v24 . Hence, R?2 has one generator of degree 8, say a, and the map d : R?2 ! R?1 is injective and sends a to v2 v4 v13 ? v1 v3 v24 . Thus, we have the minimal resolution 0 ? ?? R?2 ? ?? R?1 ? ?? R0 ? ?? M ? ?? 0; ?! ?! ?! ?! ?! 0 = 0;0 (k(K 1 )) = 1, rank R?1 = ?1;4 (k(K 1 )) = 2, rank R?2 = where rank R ?2;8 (k(K 1 )) = 1. The Betti numbers ?i;2j (k(K )) are important combinatorial invariants of simplicial complex K , see 128]. The following theorem (which was proved by combinatorial methods) reduces the calculation of ?i;2j (k(K )) to calculating the homology groups of subcomplexes of K . Theorem 3.27 (Hochster 76] or 128, Theorem 4.8]). We have ?i;2j ?k(K ) = X dimk Hj?i?1 (K ); e
where K is the ful l subcomplex of K corresponding to , see (2.1). We assume e H?1 (?) = k above. Example 3.28. Again, let K 1 be the boundary of a square, so m = 4. This time we calculate the Betti numbers ?i;2j (k(K )) using Hochster's theorem. Among two-element subsets of m] there are four simplices and two non-simplices, namely, f1; 3g and f2; 4g. Simplices contribute trivially to the sum for ?1;4 (k(K )), while each of the two non-simplices contributes 1, hence, ?1;4 (k(K )) = 2. Further, each of the four full subcomplexes corresponding to three-element subsets of m] is contractible, hence, its reduced homology vanishes and ?i;6 (k(K )) = 0 for any i. Finally, the full subcomplex K with j j = 4 is K itself, hence ?i;8 (k(K )) = e dimk H4?i?1 (K ). The latter equals 1 for i = 2 and zero otherwise. In chapter 7 we show that ?i;2j (k(K )) equals the corresponding bigraded Betti number of the moment-angle complex ZK associated to simplicial complex K . This provides an alternative (topological) way for calculating the numbers ?i;2j (k(K )). Now we turn to the Koszul resolution (Example 3.24).
m]: j j=j


44

3. COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

Tork v1 ;::: ;vm ] (M ; k) = H u1 ; : : : ; um ] M ; d ; where H u1; : : : ; um ] M ; d] is the cohomology of the bigraded di erential module u1; : : : ; um] M and d is de ned as in (3.4). Proof. Using the Koszul resolution u1 ; : : : ; um ] k v1 ; : : : ; vm ]; d] in the de nition of Tork v1 ;::: ;vm ] (k; M ), we calculate Tork v1 ;::: ;vm ] (M ; k) = Tork v1 ;::: ;vm ] (k; M ) = H u1; : : : ; um] k v1 ; : : : ; vm ] k v1;::: ;vm ] M = H u1 ; : : : ; um ] M :
Corollary 3.30. Suppose that a k v1 ; : : : ; vm ]-module M is an algebra, then Tork v1 ;::: ;vm ] (M ; k) is canonical ly a nite-dimensional bigraded k-algebra. Proof. It is easy to see that in this case the tensor product u1; : : : ; um] M is a di erential algebra, and Tork v1 ;::: ;vm ] (M ; k) is its cohomology by Lemma 3.29. Definition 3.31. The bigraded algebra Tork v1 ;::: ;vm ] (M ; k) is called the Toralgebra of algebra M . If M = k(K ) then it is called the Tor-algebra of simplicial complex K . Remark. For general N 6= k the module Tork v1 ;::: ;vm ] (M ; N ) has no canonical multiplicative structure even if both M and N are algebras. Lemma 3.32. A simplicial map : K1 ! K2 between two simplicial complexes on the vertex sets m1 ] and m2 ] respectively induces a homomorphism ? ? (3.7) t : Tork w1 ;::: ;wm2 ] k(K2 ); k ! Tork v1 ;::: ;vm1 ] k(K1 ); k of the corresponding Tor-algebras. Proof. This follows directly from Propositions 3.4 and 3.25 (b). Construction 3.33 (multigraded structure in the Tor-algebra). We may invest the polynomial ring k v1 ; : : : ; vm ] with a multigrading (more precisely, N m grading) by setting mdeg vi = (0; : : : ; 0; 2; 0; : : : ; 0) where 2 stands at the i-th i im place. Then the multidegree of monomial v11 vm is (2i1; : : : ; 2im). Suppose that algebra M is a quotient of the polynomial ring by a monomial ideal. Then the multigraded structure descends to M and to the terms of resolution (3.2). We may assume that the di erentials in the resolution preserve the multidegrees. Then the module Tork v1 ;::: ;vm ] (M ; N ) acquires a canonical N N m -grading, i.e. M Tork v1 ;::: ;vm ] (M ; k) = Tor?i;1 ;::: ;vm ] (M ; k): kv
j

Lemma 3.29. For any module M it holds that

In particular, the Tor-algebra of K is canonically an N N m -graded algebra. Remark. According to our agreement, the rst degree in the Tor-algebra is non-positive . (Remember that we numerated the terms of k v1 ; : : : ; vm ]-free Koszul resolution of k by non-positive integers.) In these notations the Koszul complex M u1; : : : ; um ]; d] becomes a cochain complex, and Tork v1 ;::: ;vm ] (M ; k) is its cohomology , not the homology as usually regarded. One of the reasons for such an

i>0; j 2Nm


3.4. HOMOLOGICAL PROPERTIES OF FACE RINGS

45

agreement is that Tork v1 ;::: ;vm ] (k(K ); k) is a contravariant functor from the category of simplicial complexes and simplicial maps, see Lemma 3.32. It also explains our notation Tork; v1 ;::: ;vm ] (M ; k), used instead of the usual Tork; v1 ;::: ;vm ](M ; k). These notations are convenient for working with Eilenberg{Moore spectral sequences, see section 7.1. The upper bound hd M 6 m from the Hilbert syzygy theorem can be replaced by the following sharper result. Theorem 3.34 (Auslander and Buchsbaum). hd M = m ? depth M . In particular, if M = k(K n?1 ) and K n?1 is Cohen{Macaulay (see De nition 3.14), then hd k(K ) = m ? n and Tor?iv1 ;::: ;vm ] (k(K ); k) = 0 for i > m ? n. k From now on we assume that M is generated by degree-two elements and the k v1 ; : : : ; vm ]-module structure in M is de ned through an epimorphism p : k v1 ; : : : ; vm] ! M (both assumptions are satis ed by de nition for M = k(K )). Suppose that 1 ; : : : ; k is a regular sequence of degree-two elements of M . Let J := ( 1 ; : : : ; k ) M be the ideal generated by 1 ; : : : ; k . Choose degree-two elements ti 2 k v1 ; : : : ; vm ] such that p(ti ) = i , i = 1; : : : ; k. The ideal in k v1 ; : : : ; vm ] generated by t1 ; : : : ; tk will be also denoted by J . Then we have k v1 ; : : : ; vm ]=J = k w1 ; : : : ; wm?k ]. Under these assumptions we have the following reduction lemma. Lemma 3.35. The fol lowing isomorphism of algebras holds for any ideal J generated by a regular sequence: Tork v1 ;::: ;vm ] (M ; k) = Tork v1 ;::: ;vm ]=J (M =J ; k): In order to prove the lemma we need the following fact from homological algeTheorem 3.36 (

bra.

Er ) Tor (A; C ); E2 = Tor A; Tor? (C; k Proof of Lemma 3.35. Set = k v1 ; : : : ; vm ], ? = k C = M . Then is a free ?-module and = ==? = k v1 ; : : :
Theorem 3.36 gives a spectral sequence

that

==? the quotient algebra. Suppose that is a free ?-module and we are given an -module A and a -module C . Then there exists a spectral sequence fEr ; dr g such
?

40, p. 349]). Let be an algebra, ? its subalgebra, and =
):

t1 ; : : : ; tk ], A = k, ; vm ]=J . Therefore,

Er ) Tork v1 ;::: ;vm ] (M ; k); E2 Since 1 ; : : : ; k is a regular sequence, M is a Tor? (M ; k) = M ? k = M =J and p;q It follows that E2 = 0 for q 6= 0. Thus, the
term, and we have
?

= Tor Tor? (M ; k); k : free ?-module. Therefore, Torq (M ; k) = 0 for q 6= 0: ? spectral sequence collapses at the E
2

?

Tork v1 ;::: ;vm ] (M ; k) = Tor Tor? (M ; k); k = Tork v1 ;::: ;vm ]=J (M =J ; k); which concludes the proof.


46

3. COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES

It follows from Theorem 3.34 that if M is Cohen{Macaulay of Krull dimension n, then depth M = n, hd M = m ? n, and Tor?iv1 ;::: ;vm ] (M ; k) = 0 for k i > m ? n. Definition 3.37. Suppose M is a Cohen{Macaulay algebra of Krull dimen? ?) sion n. Then M is called a Gorenstein algebra if Tork (vm;:::n;vm ] (M ; k) = k. 1 Following Stanley 128], we call a simplicial complex K Gorenstein if k(K ) is a Gorenstein algebra. Further, K is called Gorenstein* if k(K ) is Gorenstein and K = core K (see De nition 2.24). The following theorem characterizes Gorenstein* simplicial complexes. Theorem 3.38 ( 128, xII.5]). A simplicial complex K is Gorenstein* over k if and only if for any simplex 2 K (including = ?) the subcomplex link has the homology of a sphere of dimension dim(link ). In particular, simplicial spheres and simplicial homology spheres (triangulated manifolds with the homology of a sphere) are Gorenstein* complexes. However, the Gorenstein* property does not guarantee a complex to be a triangulated manifold (links of vertices are not necessarily simply connected, compare with Theorem 2.36). The Poincare series of the Tor-algebra and the face ring of a Gorenstein* complex are \self dual" in the following sense. Theorem 3.39 ( 128, xII.5]). Suppose K n?1 is a Gorenstein* complex. Then the fol lowing identities hold for the Poincare series of Tor?iv1 ;::: ;vm ] (k(K ); k), 0 6 k i 6 m ? n: ? ? )+ ? F Tor?iv1 ;::: ;vm ] k(K ); k ; t = t2m F Tor?(vm;:::n;vmi] k(K ); k ; 1 : k k1 t Equivalently, ?i;2j ?k(K ) = ?(m?n)+i;2(m?j) ?k(K ) ; 0 6 i 6 m ? n; 0 6 j 6 m: Corollary 3.40. If K n?1 is Gorenstein* then ? ? F k(K ); t = (?1)n F k(K ); 1 : t Proof. We apply Theorem 3.21 to a minimal resolution of k(K ). It follows from (3.6)?that the numerators of the summands in the right hand side of (3.3) are exactly F Tor?iv1 ;::: ;vm ] (k(K ); k); t , i = 1; : : : ; m ? n. Hence, k

3.5. Gorenstein complexes and Dehn{Sommerville equations

F k(K ); t = (1 ? t
? F k(K ); t = (1 ? t2 )?m

?

Using Theorem 3.39, we calculate
m?n X

m?n 2 )?m X ( i=0

?1)i F Tor? k

? i v1 ;::: ;vm ] k(K );

k ;t :

= 1?

?

i=0 m?n X ( 1 )2 ?m (?1)m (?1)m?n?j t j =0

? )+ ? (?1)i t2m F Tor?(vm;:::n;vmi] k(K ); k ; 1 k1 t
? F Tor?jv1 ;::: ;vm ] k(K ); k ; 1 k t ? = (?1)n F k(K ); 1 : t


3.5. GORENSTEIN COMPLEXES AND DEHN{SOMMERVILLE EQUATIONS

47

Corollary 3.41. The Dehn{Sommervil le relations hi = hn?i , 0 6 i 6 n, hold for any Gorenstein* complex K n?1 (in particular, for any simplicial sphere). Proof. This follows from Lemma 3.8 and Corollary 3.40. As it was pointed out by Stanley in 127], Gorenstein* complexes are the most appropriate candidates for generalizing the g-theorem to. (As we have seen, polytopal spheres, P L spheres, simplicial spheres and simplicial homology spheres are examples of Gorenstein* complexes.) The Dehn{Sommerville equations can be generalized even beyond Gorenstein* complexes. In 86] Klee reproved the f -vector version (1.10) of the Dehn{Sommerville equations in the more general context of Eulerian complexes . (A pure simplicial complex K n?1 is called Eulerian if for any simplex 2 K , including ?, holds (link ) = (S n?j j?1 ) = 1 + (?1)n?j j?1 .) Generalizations of equations (1.10) were obtained by Bayer and Billera 17] (for Eulerian posets ) and Chen and Yan 42] (for arbitrary polyhedra). In section 7.6 we deduce the generalized Dehn{Sommerville equations for triangulated manifolds as a consequence of the bigraded Poincare duality for momentangle complexes. In particular, this gives the following short form of the equations in terms of the h-vector: ? ? hn?i ? hi = (?1)i (K n?1 ) ? (S n?1 ) n ; i = 0; 1; : : : ; n: i Here (K n?1 ) = f0 ? f1 + : : : + (?1)n?1 fn?1 = 1 + (?1)n?1 hn is the Euler characteristic of K n?1 and (S n?1 ) = 1 + (?1)n?1 is that of a sphere. Note that the above equations reduce to the classical hn?i = hi in the case when K is a simplicial sphere or has odd dimension.


48

3. COMMUTATIVE AND HOMOLOGICAL ALGEBRA OF SIMPLICIAL COMPLEXES


CHAPTER 4

Cubical complexes
At some stage of development of the combinatorial topology, cubical complexes were considered as an alternative to triangulations, a new way to study topological invariants combinatorially. Later it turned out, however, that the cubical (co)homomology itself is not very advantageous in comparison with the simplicial one. Nevertheless, as we see below, cubical complexes as particular combinatorial structures are very helpful in di erent geometrical and topological considerations. A q-dimensional topological cube as a q-ball with a face structure de ned by a homeomorphism with the standard q-cube I q . A face of a topological q-cube is thus the homeomorphic image of a face of I q . Definition 4.1. A ( nite topological) cubical complex is a subset C Rn represented as a nite union U of topological cubes of any dimensions, called faces , in such a way that the following two conditions are satis ed: (a) Each face of a cube in U belongs to U ; (b) The intersection of any two cubes in U is a face of each. The dimension of C is the maximal dimension of its faces. The f -vector of a cubical complex C is f (C ) = (f0 ; f1 ; : : : ), where fi is the number of i-faces. Remark. The above de nition of cubical complex is a weaker cubical version of De nition 2.2 of geometrical simplicial complex. If we replace \topological cubes" in De nition 4.1 by \convex polytopes combinatorially equivalent to I q ", then we get the de nition of a combinatorial-geometrical cubical complex , or cubical polyhedron . One can also de ne an abstract cubical complex as a poset (more precisely, a semilattice) such that each interval 0; t] is isomorphic to the face lattice of a cube. We would not discuss here relationships between topological, geometrical and abstract cubical complexes, since all examples we need for our further constructions constitute a rather restricted family. The theory of f -vectors of cubical complexes is parallel, to a certain extent, to that of simplicial complexes, but is much less developed. It includes the notions of h-vector, Cohen{Macaulay and Gorenstein* cubical complexes, and there are cubical analogues of the UBC, LBC and g-conjecture. See 3] and 10] for more details. A brief review of this theory and references can be found in 129, x2]. Since the combinatorial theory of cubical complexes is still in the early stage of its development, it may be helpful to look at some possible applications. It turns out that some particular problems from the discrete geometry and combinatorics of cubical complexes arise naturally in statistical physics, namely in connection with the 3-dimensional Ising model. Since this aspect is not widely known to combinatorialists, we make a brief digression to the corresponding problems.
49

4.1. De nitions and cubical maps


50

4. CUBICAL COMPLEXES

The standard unit cube I q = 0; 1]q , together with all its faces, is a q-dimensional cubical complex, which we will also denote I q . Unlike simplicial complexes, which are always realizable as subcomplexes in a simplex, not every cubical complex appears as a subcomplex of some I q . One example of a cubical complex not embeddable as a subcomplex in any I q is shown on Figure 4.1. Moreover, this complex is not embeddable into the standard cubical lattice in Rq (for any q). The authors are thankful to M. I. Shtogrin for presenting this example.
@ C C C HH H @ HH @ H @ ?@ @? @ @? @ ?@ ? @ ? @? A ? A A ? A? A A

@ @ XXX A A

A A ?H ?H A? HH H

Figure 4.1. Cubical complex not embeddable into cubical lattice.

(a) a (cubical) embedding into the standard cubical lattice in Rq ; (b) a map to the standard cubical lattice in Rq whose restriction to every kdimensional cube is an isomorphism with a certain k-face of the lattice. In the case when C is homeomorphic to a 2-sphere the above problem was solved in 57]. Problem 4.2 is an extension of the following question, also formulated in 57]. Problem 4.3 (S. P. Novikov). Suppose we are given a 2-dimensional cubical mod 2 cycle in the standard cubical lattice in R3 . Describe al l maps of cubical subdivisions of 2-dimensional surfaces onto such that no two di erent squares are mapped to the same square of . As it was told to the authors by S. P. Novikov, the above question was raised during his discussions with A. M. Polyakov on the 3-dimensional Ising model. Here we introduce some particular cubical complexes, which will play a pivotal r^ in our further constructions (in particular in the theory of moment-angle ole complexes). We make no claims for originality of constructions appearing in this section | most of them are part of mathematical folklore. At the end we give several references to the sources where some similar considerations can be found. All cubical complexes discussed here admit a canonical cubical embedding into the standard cube. To conclude the discussion in the end of the previous section we note that the problem of embeddability into the cubical lattice is closely connected with that of embeddability into the standard cube. For instance, it is shown in 57]

C (in particular, cubical manifolds) which admit

Problem 4.2 (S. P. Novikov). Characterize k-dimensional cubical complexes

4.2. Cubical subdivisions of simple polytopes and simplicial complexes


4.2. CUBICAL SUBDIVISIONS OF POLYTOPES AND SIMPLICIAL COMPLEXES

51

cubical subdivision of a 2-dimensional surface is embeddable into the cubical lattice in Rq , then it also admits a cubical embedding into I q . face of I q can be written as C = f(y1 ; : : : ; yq ) 2 I q : yi = 0 for i 2 ; yi = 1 for i 2 g; = are two (possibly empty) subsets of q]. We set C := C? . Construction 4.4 (canonical simplicial subdivision of I m ). Let = m?1 be the simplex on the set m], i.e. the collection of all subsets of m]. Assign to each subset = fi1; : : : ; ik g m] the vertex v := C of I m . More explicitly, v = ("1 ; : : : ; "m ), where "i = 0 if i 2 and "i = 1 otherwise. Regarding each as a vertex of the barycentric subdivision of , we can extend the correspondence 7! v to a piecewise linear embedding of the barycentric subdivision 0 into the (boundary complex of ) standard cube I m . Under this embedding, denoted ic , the vertices of are mapped to the vertices of I m having only one zero coordinate, while the barycenter of is mapped to (0; : : : ; 0) 2 I m (see Figure 4.2). The image ic( 0 ) is the union of m facets of I m meeting at the vertex (0; : : : ; 0). For each pair of non-empty subsets of m] all simplices of 0 of the form = 1 2 k= are mapped to the same face C I m . The map ic : 0 ! I m extends to 0 ) by taking the vertex of the cone to (1; : : : ; 1) 2 I m . We denote the cone( resulting map by cone(ic ). Its image is the whole I m . Hence, cone(ic) : cone( 0 ) ! I m is a P L homeomorphism linear on the simplices of cone( 0 ). This de nes a triangulation of I m which coincides with the canonical triangulation of the product of m one-dimensional simplices, see Construction 2.11. It is also known as the \standard triangulation along the main diagonal". In short, it can be said that the canonical triangulation of I m arises from the identi cation of I m with the cone over the barycentric subdivision of m?1 . Construction 4.5 (cubical subdivision of a simple polytope). Let P n Rn n be a simple polytope with m facets F1n?1 ; : : : ; Fm?1 . Choose a point in the relative n , including the vertices and the polytope itself. We get interior of every face of P the set S of 1 + f0 + f1 + : : : + fn?1 points (here f (P n ) = (f0 ; f1; : : : ; fn?1 ) is the f -vector of P n ). For each vertex v 2 P n de ne the subset Sv S consisting of the points chosen inside?the faces containing v. Since P n is simple, the number of k-faces meeting at v is n , 0 6 k 6 n. Hence, jSv j = 2n . The set Sv will be the k n n vertex set of an n-cube, which we denote Cv . The faces of Cv can be described as k and Gl be two faces of P n such that v 2 Gk Gl . Then there are follows. Let G1 2 1 2 exactly 2l?k faces of P n between Gk and Gl2 . The corresponding 2l?k points from 1 l n S form the vertex set of an (l ? k)-face of Cv . We denote this face CG?k G2 . Every 1 n i face of Cv is CG1 G2 for some G1 ; G2 containing v. The intersection of any two nn cubes Cv , Cv0 is a face of each. Indeed, let Gi P n be the smallest face containing ni n n n n both vertices v and v0 . Then Cv \ Cv0 = CG?i P n is the face of both Iv and Iv0 . n with fn?1 (P n ) cubes of Thereby we have constructed a cubical subdivision of P dimension n. We denote this cubical complex by C (P n ). There is an embedding of C (P n ) to I m constructed as follows. Every (n ? k)face of P n is the intersection of k facets: Gn?k = Fin?1 \ : : : \ Fin?1 . We map 1 k the corresponding point of S to the vertex ("1 ; : : : ; "m) 2 I m where "i = 0 if i 2 fi1 ; : : : ; ik g and "i = 1 otherwise. This de nes a mapping from the vertex set S of C (P n ) to the vertex set of I m . Using the canonical triangulation of I m from Construction 4.4, we extend this mapping to a P L embedding iP : P n ! I m . For

that if a standard Any (4.1) where


52

4. CUBICAL COMPLEXES

Q AQ AQ A QQ A Q Q A Q 123 Q A Q A Q A Q Q 23A 13 A A A A A A @ ic A @ R @ 3A

2

12

1

@ @

ic(12)
@ @ vertex of @ cone @ ?@ ? @ ? @ ? @

ic(1)

ic (2)
? ? ?

@ @

ic (23)

? ?

?

?

?

?

?

?

@

i

@ @

c (123)

ic(13)

ic (3)

Figure 4.2. Cone over 0 as the standard triangulation of cube.

each vertex v = Fin?1 \ \ Fin?1 2 P n we have 1 n n ) = (y1 ; : : : ; ym ) 2 I m : yj = 1 for j 2 fi1 ; : : : ; in g ; (4.2) iP (Cv = n i.e. iP (Cv ) = Cfi1 ;::: ;in g I m in the notation of (4.1). The embedding iP : P n ! m for n = 2, m = 3 is shown in Figure 4.3. I We summarize the facts from the above construction in the following statement. Proposition 4.6. A simple polytope P n with m facets can be split into cubes n , one for each vertex v 2 P n . The resulting cubical complex C (P n ) embeds Cv canonical ly into the boundary of I m , as described by (4.2).


4.2. CUBICAL SUBDIVISIONS OF POLYTOPES AND SIMPLICIAL COMPLEXES
P

53

n

C
A A A A A

I

m
?

C
? ? ? ? ?

D G E

B
AD A A A

B

HH H

iP
A A A

? ?

G

0

A

F

E

A

F

?

? ?

Figure 4.3. The embedding iP : P n

! I m for n = 2, m = 3.

Construction 4.8. Let K n?1 be a simplicial complex on m]. Then K is naturally a subcomplex of m?1 and K 0 is a subcomplex of ( m?1 )0 . As it follows from Construction 4.4, there is a P L embedding icjK 0 : jK 0 j ! I m . The image ic(jK 0 j) is an (n ? 1)-dimensional cubical subcomplex of I m , which we denote cub(K ). We

Lemma 4.7. The number of k-faces of the cubical complex C (P n ) is given by n?k ? X ? n?i fn?i?1 (P n ) fk C (P n ) = k i=0 ? ?n = k fn?1 (P n ) + n?1 fn?2 (P n ) + + fk?1 (P n ); k = 0; : : : ; n: k Proof. This follows from the fact that the k-faces of C (P n ) are in one-to-one correspondence with the pairs Gi Gi+k of embedded faces of P n .
1 2

have (4.3)

cub(K ) =

i.e. cub(K ) is the union of faces C I m over all pairs of non-empty simplices of K . Construction 4.9. Since cone(K 0 ) is a subcomplex of cone(( m?1 )0 ), Construction 4.4 also provides a P L embedding cone(ic)jcone(K 0 ) : j cone(K 0 )j ! I m : The image of this embedding is an n-dimensional cubical subcomplex of I m , which we denote cc(K ). It can be easily seen that (4.4) cc(K ) = C=C (the latter identity holds since C C? = C ). Remark. If fig 2 m] is not a vertex of K , then cc(K ) is contained in the facet fyi = 1g of I m .
2K 2K

?= 6

2K

C

I m;


54

4. CUBICAL COMPLEXES

The following statement summarizes the results of two previous constructions. Proposition 4.10. For any simplicial complex K on the set m] there is a P L embedding of the polyhedron jK j into I m linear on the simplices of K 0 . The image of this embedding is the cubical subcomplex (4.3). Moreover, there is a P L embedding of the polyhedron j cone(K )j into I m linear on the simplices of cone(K 0 ), whose image is the cubical subcomplex (4.4). A cubical complex C 0 is called a cubical subdivision of cubical complex C if each cube of C is a union of nitely many cubes of C 0. Proposition 4.11. For every cubical subcomplex C there exists a cubical subdivision that is embeddable into some I q as a subcomplex. Proof. Subdividing each cube of C as described in Construction 4.4 we obtain a simplicial complex, say KC . Then applying Construction 4.8 to KC we get a cubical complex cub(KC ) that subdivides KC and therefore C . It is embeddable into some I q by Proposition 4.10.

u u

? ?

? ?

? ?

u u

? ?

? ?

0

(a) K = 3 points

(b) K = @ Figure 4.4. The cubical complex cub(K ).

u u u

?

? ?

u u u u

? ? ?? ? ? ? ?

u u

? ? ?

? ?

? ?

0

u

? ? ? ? ? ?

u u

2

? ?

?

? ?

? ? ? ?

? ?

? ?

0

?

? ?

?? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ?? ? ? ? ? ? ? ? ??? ??? ??? 0 ??? ? ??? ? ? ? ?? ? ? ? ?

(a) K = 3 points

(b) K = @ Figure 4.5. The cubical complex cc(K ).

u u u u
2

Example 4.12. The cubical complex cub(K ) in the case when K is a disjoint union of 3 vertices is shown in Figure 4.4 (a). Figure 4.4 (b) shows that for the case K = @ 2 , the boundary complex of a 2-simplex. The corresponding cubical complexes cc(K ) are indicated in Figure 4.5 (a) and (b).


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55

Remark. As a topological space, cub(K ) is homeomorphic to jK j, while cc(K ) is homeomorphic to j cone(K )j. On the other hand, there is the cubical complex cub(cone(K )), also homeomorphic to j cone(K )j. However, as cubical complexes , cc(K ) and cub(cone(K )) di er (since cone(K 0 ) 6= (cone(K ))0 ). Let P be a simple n-polytope and KP the corresponding simplicial (n ? 1)sphere (the boundary of the polar simplicial polytope P ). Then cc(KP ) coincides with the cubical complex C (P ) from Construction 4.5. More precisely, cc(KP ) = iP (C (P )). Thus, Construction 4.5 is a particular case of Construction 4.9 (compare Figures 4.2{4.5). Remark. Di erent versions of Construction 4.9 can be found in 10] and in some earlier papers listed there on p. 299. In 48, p. 434] a similar construction was introduced while studying certain toric spaces; we will return to this in the next chapters. A version of the cubical subcomplex cub(K ) I m appeared in 120] in connection with Problem 4.2.


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CHAPTER 5

Toric and quasitoric manifolds
Toric varieties appeared in algebraic geometry in the beginning of 1970s in connection with compacti cation problems for algebraic torus actions. The geometry of toric varieties very quickly has become one of the most fascinating topics in algebraic geometry and found applications in many mathematical sciences, which otherwise seemed far from each other. We have already mentioned the proof for the \only if" part of the g-theorem for simplicial polytopes given by Stanley. Other remarkable applications include counting lattice points and volumes of lattice polytopes; relations with Newton polytopes and singularities (after Khovanskii and Kushnirenko); discriminants, resultants and hypergeometric functions (after Gelfand, Kapranov and Zelevinsky); re exive polytopes and mirror symmetry for Calabi{Yau toric hypersurfaces (after Batyrev). Standard references in the toric geometry are Danilov's survey 46] and books by Oda 105], Fulton 64] and Ewald 61]. A more recent survey article by Cox 45] covers new applications, including those mentioned above. We are not going to give another review of the toric geometry here. Instead, in this section we stress upon some topological and combinatorial aspects of toric varieties. We also give Stanley's argument for the g-theorem.

5.1. Toric varieties

5.1.1. Toric varieties and fans. Let C = C n f0g denote the multiplicative group of complex numbers. The product (C )n of n copies of C is known as the torus in the theory of algebraic groups. In topology, the torus T n is the product of n circles. We keep the topological notations, referring to (C )n as the algebraic torus . The torus T n is a subgroup of the algebraic torus (C )n in the standard way: ? (5.1) T n = e2 i'1 ; : : : ; e2 i'n 2 C n ; where ('1 ; : : : ; 'n ) is running through Rn . Definition 5.1. A toric variety is a normal algebraic variety M containing the algebraic torus (C )n as a Zariski open subset in such a way that the natural action of (C )n on itself extends to an action on M . Hence, (C )n acts on M with a dense orbit. One of the most beautiful properties of toric varieties is that all of their subtlest algebro-geometrical properties can be translated into the language of combinatorics and convex geometry. The following de nition introduces necessary combinatorial notions. Definition 5.2 (Fans terminology). Let Rn be the Euclidean space and Zn n the integral lattice. Given a nite set of vectors l 1 ; : : : ; l s 2 Rn , de ne the R
57


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convex polyhedral cone spanned by l 1 ; : : : ; l s by (5.2) = fr1 l 1 + + rs l s 2 Rn : ri > 0g: Any convex polyhedral cone is a convex polyhedron in the sense of De nition 1.2. Hence, the faces of a convex polyhedral cone are de ned. A cone is rational if its generator vectors l 1 ; : : : ; l s can be taken from Zn and is strongly convex if it contains no line through the origin. All cones considered below are strongly convex and rational. A cone is simplicial (respectively, non-singular ) if it is generated by a part of a basis of Rn (respectively, Zn). A fan is a set of cones in Rn such that each face of a cone in is also a cone in , and the intersection of two cones in is a face of each. A fan in Rn is called complete if the union of all cones from is Rn . A fan is simplicial (respectively, non-singular ) if all cones of are simplicial (respectively, non-singular). Let be a simplicial fan in Rn with m one-dimensional cones (or rays ). Choose generator vectors l 1 ; : : : ; l m for these rays to be integer and primitive, i.e. with relatively prime integer coordinates. The fan de nes a simplicial complex K on the vertex set m], which is called the underlying complex of . By de nition, fi1 ; : : : ; ik g m] is a simplex of K if and only if l i1 ; : : : ; l ik span a cone of . Obviously, is complete if and only if K is a simplicial (n ? 1)-sphere. As it is explained in any of the above mentioned sources, there is a one-to-one correspondence between fans in Rn and toric varieties of complex dimension n. We will denote the toric variety corresponding to a fan by M . It follows that, in principle, all geometrical and topological properties of a toric variety can be retrieved from the combinatorics of the underlying fan. The inclusion poset of (C )n -orbits of M is isomorphic to the poset of faces of with reversed inclusion. That is, the k-dimensional cones of correspond to the codimension-k orbits of the algebraic torus action on M . In particular, the n-dimensional cones correspond to the xed points, while the origin corresponds to the unique dense orbit. The toric variety M is compact if and only if is complete. If is simplicial then M is an orbifold (i.e. is locally homeomorphic to the quotient of R2n by a nite group action). Finally, M is non-singular (smooth) if and only if is non-singular, which explains the notation. Smooth toric varieties sometimes are called toric manifolds in the algebraic geometry literature. Remark. Bistellar moves (see De nition 2.39) on the simplicial complex K can be interpreted as operations on the fan . On the level of toric varieties, such an operation corresponds to a ip (a blow-up followed by a subsequent blow-down along di erent subvariety). This issue is connected with the question of factorization of a proper birational morphism between two complete smooth (or normal) algebraic varieties of dimension > 3 into a sequence of blow-ups and blow-downs with smooth centers, a fundamental problem in the birational algebraic geometry. Two versions of this problem are usually distinguished: the Strong factorization conjecture, which asks if it is possible to represent a birational morphism by a sequence of blow-ups followed by a sequence of blow-downs, and the Weak factorization conjecture, in which the order of blow-ups and blow-downs is insigni cant. Since all toric varieties are rational, any two toric varieties of the same dimension are birationally equivalent. Weak (equivariant) factorization conjecture for smooth complete toric varieties was proved by Wlodarczyk 140] (announced in 1991) using interpretation of equivariant ips on toric varieties as bistellar move-type operations


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on the corresponding fans. Thereby the weak factorization theorem for smooth toric varieties reduces to the statement that any two complete non-singular fans in Rn can be taken one to another by a nite sequence of bistellar move-type operations in which all intermediate fans are non-singular. This result is the essence of 140]. (Note that the statement does not reduce to Pachner's Theorem 2.40 because of the additional smoothness condition.) The equivariant toric strong factorization conjecture was proved by Morelli 102]. 5.1.2. Cohomology of non-singular toric varieties. The Danilov{Jurkiewicz theorem allows us to read the integer cohomology ring of a non-singular toric variety directly from the underlying fan . Write the primitive integer vectors along the rays of in the standard basis of Zn: l j = (l1j ; : : : ; lnj )t ; j = 1; : : : ; m: Assign to each vector l j the indeterminate vj of degree 2, and de ne linear forms i := li1 v1 + + lim vm 2 Z v1; : : : ; vm ]; 1 6 i 6 n: Denote by J the ideal in Z v1; : : : ; vm ] spanned by these linear forms, i.e. J = ( 1 ; : : : ; n ). The images of 1 ; : : : ; n and J in the Stanley{Reisner ring Z(K ) = Z v1; : : : ; vm ]=IK (see De nition 3.1) will be denoted by the same symbols. Theorem 5.3 (Danilov and Jurkiewicz). Let be a complete non-singular fan in Rn , and M the corresponding toric variety. Then (a) The Betti numbers (the ranks of homology groups) of M vanish in odd dimensions, while in even dimensions are given by b2i (M ) = hi (K ); i = 0; 1; : : : ; n; where h(K ) = (h0 ; : : : ; hn ) is the h-vector of K . (b) The cohomology ring of M is given by H (M ; Z) = Z v1; : : : ; vm ]=(IK + J ) = Z(K )=J ; where vi , 1 6 i 6 m, denote the 2-dimensional cohomology classes dual to invariant divisors (codimension-two submanifolds) Di corresponding to the rays of . Moreover, 1 ; : : : ; n is a regular sequence in Z(K ). This theorem was proved by Jurkiewicz 82] for pro jective smooth toric varieties and by Danilov 46, Theorem 10.8] in the general case. Note that the ideal IK is determined only by the combinatorics of the fan (i.e. by the intersection poset of ), while J depends on the geometry of . One can observe that the rst part of Theorem 5.3 follows from the second part and Lemma 3.8. Remark. As it was shown by Danilov, the Q -coe cient version of Theorem 5.3 is also true for simplicial fans and toric varieties. It follows from Theorem 5.3 that the cohomology of M is generated by twodimensional classes. This is the rst thing to check if one wishes to determine whether or not a given algebraic variety or smooth manifold arises as a non-singular (or simplicial) toric variety. Another interesting algebraic-geometrical property of non-singular toric varieties, suggested by Theorem 5.3, is that the Chow ring 64, x 5.1] of M coincides with its integer cohomology ring.


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Construction 5.4 (Normal fan and toric varieties from polytopes). Suppose we are given an n-polytope (1.1) with vertices in the integer lattice Zn Rn . Such a polytope is called integral, or lattice . Then the vectors l i in (1.1), 1 6 i 6 m, can be chosen integer and primitive, and the numbers ai can be chosen integer. Note that l i is normal to the facet Fi P n and is pointing inside the polytope P . De ne the complete fan (P ) whose cones are generated by those sets of normal vectors l i1 ; : : : ; l ik whose corresponding facets Fi1 ; : : : ; Fik have non-empty intersection in P . The fan (P ) is called the normal fan of P . Alternatively, if 0 2 P then the normal fan consists of cones over the faces of the polar polytope P . De ne the toric variety MP := M (P ) . The variety MP is smooth if and only if P is simple and the normal vectors l i1 ; : : : ; l in of any set of n facets Fi1 ; : : : ; Fin meeting at the same vertex form a basis of Zn. Remark. Every combinatorial simple polytope is rational , that is, admits a convex realization with rational vertex coordinates. Indeed, there is a small perturbation of de ning inequalities in (1.1) that makes all of them rational but does not change the combinatorial type (since the half-spaces de ned by the inequalities are in general position). As a result, one gets a simple polytope P 0 of the same combinatorial type with rational vertex coordinates. To obtain a realization with integral vertex coordinates we just take the magni ed polytope kP 0 for appropriate k 2 Z. We note that this is not the case in general: in every dimension > 5 there exist non-rational convex polytopes (non-simple and non-simplicial), see e.g. 145, Example 6.21] and discussion there. In dimension 3 all convex polytopes are rational, and in dimension 4 the existence of non-rational polytopes is an open problem. Returning to simple polytopes, we note that di erent realizations of a given combinatorial simple polytope as lattice polytopes may produce di erent (even topologically) toric varieties MP . At the same time there exist combinatorial simple polytopes that do not admit any lattice realization with smooth MP . We present one such example in the next section, see Example 5.26. The underlying topological space of a toric variety MP can be identi ed with the quotient space T n P n = for some equivalence relation using the following construction (see e.g. 64, x 4.1]). Construction 5.5 (Toric variety as an identi cation space). We identify the torus T n (5.1) with the quotient Rn =Zn. For each point q 2 P n de ne G(q) as the smallest face that contains q in its relative interior. The normal subspace to G(q), denoted N , is spanned by the primitive vectors l i (see (1.1)) corresponding to those facets Fi which contain G(q). (If P n is simple then there are exactly codim G(q) such facets; in general there are more of them.) Since N is a rational subspace, it pro jects to a subtorus of T n, which we denote T (q). Note that dim T (q) = n ? dim G(q). Then, as a topological space, MP = T n P n = ; where (t1 ; p) (t2 ; q) if and only if p = q and t1 t?1 2 T (q). The subtori T (q) 2 are the isotropy subgroups for the action of T n on MP , and P n is identi ed with the orbit space. Note that if q is a vertex of P n then T (q) = T n, so the vertices correspond to the T n- xed points of MP . At the other extreme, if q 2 int P n then T (q) = feg, so the T n-action is free over the interior of the polytope. More

5.1.3. Toric varieties from polytopes.


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generally, if : MP ! P n is the quotient pro jection then ?1 ?int G(q) = ?T n=T (q) int G(q): Remark. The above construction can be generalized to all complete toric varieties (not necessarily coming from polytopes) by replacing P n by an n-ball with cellular decomposition on the boundary. This cellular decomposition is \dual" to that de ned by the complete fan. Construction 5.4 allows us to de ne the simplicial fan (P ) and the toric variety MP from any lattice simple polytope P . However, the lattice polytope P contains more geometrical information than the fan (P ). Indeed, besides the normal vectors l i , which determine the fan, we also have numbers ai 2 Z, 1 6 i 6 m, (see (1.1)). In the notations of Theorem 5.3, it is well known in the toric geometry that the linear combination D = a1 D1 + + am Dm is an ample divisor on MP . It de nes a pro jective embedding MP C P r for some r (which can be taken to be the number of vertices of P ). This implies that all toric varieties from polytopes are pro jective. Conversely, given a smooth pro jective toric variety M C P r , one gets very ample divisor (line bundle) D of a hyperplane section whose zero cohomology is generated by the sections corresponding to lattice points in a certain lattice simple polytope P . For this P one has M = MP . Let ! := a1 v1 + + am vm 2 H 2 (MP ; Q ) be the cohomology class of D. Theorem 5.6 (Hard Lefschetz theorem for toric varieties). Let P n be a lattice simple polytope (1.1), MP the toric variety de ned by P , and ! = a1 v1 + + am vm 2 H 2 (MP ; Q ) the above de ned cohomology class. Then the maps i H n?i (MP ; Q ) ? ?? H n+i (MP ; Q ); ?! ! 1 6 i 6 n; are isomorphisms. It follows from the pro jectivity that if MP is smooth then it is Kahler, and ! is the class of the Kahler 2-form. Remark. As it is stated, Theorem 5.6 applies only to simplicial pro jective toric varieties. However it remains true for any pro jective toric variety if we replace the ordinary cohomology by the (middle perversity) intersection cohomology . For more details see the discussion in 64, x 5.2]. Example 5.7. The complex pro jective space C P n = f(z0 : z1 : : zn); zi 2 C g is a toric variety. The algebraic torus (C )n acts on C P n by (t1 ; : : : ; tn ) (z0 : z1 : : zn ) = (z0 : t1 z1 : : tn zn ): Obviously, (C )n C n C P n is a dense open subset. A sample fan de ning C P n consists of the cones spanned by all proper subsets of the set of (n + 1) vectors e1 ; : : : ; en ; ?e1 ? ? en in Rn . Theorem 5.3 identi es the cohomology ring H (C P n ; Z) = Z u]=(un+1), dim u = 2, with the quotient ring Z v1; : : : ; vn+1 ]=(v1 vn+1 ; v1 ? vn+1 ; : : : ; vn ? vn+1 ): The toric variety C P n arises from a polytope: C P n = MP , where P is the standard n-simplex (1.2). The corresponding class ! 2 H 2 (C P n ; Q ) from Theorem 5.6 is represented by vn+1 . Now we are ready to give Stanley's argument for the \only if" part of the g-theorem for simple polytopes.


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Proof of the necessity part of Theorem 1.29. Realize the simple polytope as a lattice polytope P n Rn . Let MP be the corresponding toric variety. Part (a) is already proved (Theorem 1.20). It follows from Theorem 5.6 that the multiplication by ! 2 H 2 (MP ; Q ) is a monomorphism H 2i?2 (MP ; Q ) ! H 2i (MP ; Q ) for i 6 n . This together with part (a) of Theorem 5.3 gives hi?1 6 hi , 2 0 6 i 6 n , thus proving (b). To prove (c), de ne the graded commutative Q 2 algebra A := H (MP ; Q )=(!). Then A0 = Q , A2i = H 2i (MP ; Q )=! H 2i?2 (MP ; Q ) for 1 6 i 6 n , and A is generated by degree-two elements (since so is H (MP ; Q )). 2 It follows from Theorem 1.32 that the numbers dim A2i = hi ? hi?1 , 0 6 i 6 n , 2 are the components of an M -vector, thus proving (c) and the whole theorem. Remark. The Dehn{Sommerville equations now can be interpreted as the Poincare duality for MP . Even though MP needs not to be smooth, the rational cohomology algebra of a simplicial toric variety (or toric orbifold) still satis es the Poincare duality. The Hard Lefschetz theorem (Theorem 5.6) holds only for pro jective toric varieties. This implies that Stanley's argument cannot be directly generalized beyond the polytopal sphere case. So far this case is the only generality in which methods involving the Hard Lefschetz theorem are e cient for proving the g-theorem (see also the discussion at the end of section 7.6). However, the cohomology of toric varieties has been shown to be quite helpful in generalizing statements like the g-theorem in a di erent direction, namely, to the case of general (not necessarily simple or simplicial) convex polytopes. So suppose P n is a convex lattice n-polytope. It gives rise, as described in Construction 5.4, to a pro jective toric variety MP . If P n is not simple then MP has worse than just orbifold singularities and its ordinary cohomology behaves badly. The Betti numbers of MP are not determined by the combinatorial type of P n and do not satisfy the Poincare duality. On the other hand, it turns out that the dimensions b i of the intersection h cohomology of MP are combinatorial invariants of P n . The vector b h (P n ) = (b 0 ; b 1 ; : : : ; b n ) hh h is called the intersection h-vector of P n . If P n is simple, then the intersection b h-vector coincides with the ordinary one, but in general h (P n ) is not determined n and its combinatorial de nition is quite subtle, see 126] by the face vector of P for details. The intersection h-vector satis es the \Dehn{Sommerville equations" b i = b n?i , and the Hard Lefschetz theorem shows that it also satis es the GLBC hh inequalities:
b

In the case when P n cannot be realized as a lattice polytope (that is, P n is nonrational, see the remark after Construction 5.4) the combinatorial de nition of intersection h-vector still works, but it is not known whether the above inequalities continue to hold. Some progress in this direction has been achieved in 27], 134]. To summarise, we may say that although Hard Lefschetz and intersection cohomology methods so far are not very helpful in the non-convex situation (like P L or simplicial spheres), they still are quite powerful in the case of general convex polytopes. Now we look more closely at the action of the torus T n (C )n on a nonsingular compact toric variety M . This action is \locally equivalent" to the standard

h0 6 b 1 6 : : : 6 b n : h h
2


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action of T n on C n , see the next section for the precise de nition. The orbit space M =T n is homeomorphic to an n-ball, invested with the topological structure of manifold with corners by the xed point sets of appropriate subtori, see 64, x 4.1]. Roughly speaking, a manifold with corners is a space that is locally modelled by open subsets of the positive cone Rn (1.4). From this description it is easy to + deduce the strict de nition 80], which we omit here. Construction 5.8. Let P n be a simple polytope. For any vertex v 2 P n denote by Uv the open subset of P n obtained by deleting all faces not containing v. Obviously, Uv is di eomorphic to Rn (and even a nely isomorphic to an open set + of Rn containing 0). It follows that P n is a manifold with corners, with atlas fUv g. + As suggested by Construction 5.5, if smooth M arises from a lattice polytope P n (which is therefore simple) then the orbit space M =T n is di eomorphic, as a manifold with corners, to P n . Furthermore, in this case there exists an explicit map M ! Rn (the moment map ) with image P n Rn and T n-orbits as bres, see 64, x4.2]. (We will return to moment maps and some aspects of symplectic geometry in section 8.2.) The identi cation space description of a non-singular pro jective toric variety (Construction 5.5) motivated Davis and Januszkiewicz 48] to introduce a topological counterpart of the toric geometry, namely, the study of quasitoric manifolds . We proceed with their description in the next section. Quasitoric manifolds can be viewed as a \topological approximation" to algebraic non-singular pro jective toric varieties. This notion appeared in 48] under the name \toric manifolds". We use the term \quasitoric manifold", since \toric manifold" is reserved in the algebraic geometry for \non-singular toric variety". In the consequent de nitions we follow 48], taking into account adjustments and speci cations from 38]. As in the case of toric varieties, we rst give a de nition of a quasitoric manifold from the general topological point of view (as a manifold with a certain nice torus action), and then specify a combinatorial construction (similar to the construction of toric varieties from fans or polytopes).

5.2. Quasitoric manifolds

5.2.1. Quasitoric manifolds and characteristic maps. As in the previous section, we regard the torus T n as the standard subgroup (5.1) in (C )n , thereby specifying the orientation and the coordinate subgroups Ti = S 1 (i = 1; : : : ; n) in T n. We refer to the representation of T n by diagonal matrices in U (n) as the standard action on C n . The orbit space of this action is the positive cone Rn (1.4). + The canonical pro jection T n Rn ! C n : (t1 ; : : : ; tn ) (x1 ; : : : ; xn ) ! (t1 x1 ; : : : ; tn xn ) + identi es C n with a quotient space T n Rn = . This quotient will serve as the + \local model" for some other identi cation spaces below. Let M 2n be a 2n-dimensional manifold with an action of the torus T n (a T n manifold for short). Definition 5.9. A standard chart on M 2n is a triple (U; f ; ), where U is a n -stable open subset of M 2n , is an automorphism of T n , and f is a -equivariant T homeomorphism f : U ! W with some (T n -stable) open subset W C n . (The latter means that f (t y) = (t)f (y) for all t 2 T n, y 2 U .) Say that a T n-action


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on M 2n is local ly standard if M 2n has a standard atlas, that is, every point of M 2n lies in a standard chart. The orbit space for a locally standard action of T n on M 2n is an n-dimensional manifold with corners. Quasitoric manifolds correspond to the case when this orbit space is di eomorphic, as a manifold with corners, to a simple polytope P n . Note that two simple polytopes are di eomorphic as manifolds with corners if and only if they are combinatorially equivalent. Definition 5.10. Given a combinatorial simple polytope P n , a T n-manifold 2n is called a quasitoric manifold over P n if the following two conditions are M satis ed: (a) the T n-action is locally standard; (b) there is a pro jection map : M 2n ! P n constant on T n-orbits which maps every k-dimensional orbit to a point in the interior of a codimension-k face of P n , k = 0; : : : ; n. It follows that the T n -action on a quasitoric manifold M 2n is free over the interior of the quotient polytope P n , while the vertices of P n correspond to the T n- xed points of M 2n . Direct comparison with Construction 5.5 suggests that every smooth (pro jective) toric variety MP coming from a simple lattice polytope P n is a quasitoric manifold over the corresponding combinatorial polytope. We will return to this below in Example 5.19. Suppose P n has m facets F1 ; : : : ; Fm . By the de nition, for every facet Fi , the pre-image ?1 (int Fi ) consists of codimension-one orbits with the same 1dimensional isotropy subgroup, which we denote T (Fi ). It can be easily seen that ?1 (Fi ) is an 2(n ? 1)-dimensional quasitoric (sub)manifold over Fi , with respect to the action of T n=T (Fi ). We denote it Mi2(n?1) and refer to it as the facial submanifold corresponding to Fi . Its isotropy subgroup T (Fi ) can be written as ? (5.3) T (Fi ) = e2 i 1i ' ; : : : ; e2 i ni ' 2 T n ; where ' 2 R and i = ( 1i ; : : : ; ni )t 2 Zn is a primitive vector. This i is determined by T (Fi ) only up to a sign. A choice of sign speci es an orientation for T (Fi ). For now we do not care about this sign and choose it arbitrarily. A more detailed treatment of signs and orientations is the sub ject of the next section. We refer to i as the facet vector corresponding to Fi . The correspondence (5.4) ` : Fi 7! T (Fi ) is called the characteristic map of M 2n . Suppose we have a codimension-k face Gn?k , written as an intersection of k facets: Gn?k = Fi1 \ \ Fik . Then the submanifolds Mi1 ; : : : ; Mik intersect transversally in a submanifold M (G)2(n?k) , which we refer to as the facial submanifold corresponding to G. The map T (Fi1 ) T (Fik ) ! T n is injective since T (Fi1 ) T (Fik ) is identi ed with the k-dimensional isotropy subgroup of M (G)2(n?k) . It follows that the vectors i1 ; : : : ; ik form a part of an integral basis of Zn. Let be integer (n m)-matrix whose i-th column is formed by the coordinates of the facet vector i , i = 1; : : : ; m. Every vertex v 2 P n is an intersection of n facets: v = Fi1 \ \ Fin . Let (v) := (i1 ;::: ;in ) be the maximal minor of formed


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by the columns i1 ; : : : ; in . Then (5.5) det (v) = 1: The correspondence Gn?k 7! isotropy subgroup of M (G)2(n?k) extends the characteristic map (5.4) to a map from the face poset of P n to the poset of subtori of T n . Definition 5.11. Let P n be a combinatorial simple polytope and ` is a map from facets of P n to one-dimensional subgroups of T n. Then (P n ; `) is called a characteristic pair if `(Fi1 ) `(Fik ) ! T n is injective whenever Fi1 \ \ Fik 6= ?. The map ` directly extends to a map from the face poset of P n to the poset of subtori of T n, so we have subgroup `(G) T n for every face G of P n . As in the case of standard action of T n on C n , there is a pro jection T n P n ! 2n whose bre over x 2 M 2n is the isotropy subgroup of x. This argument can be M used for reconstructing the quasitoric manifold from any given characteristic pair (P n ; `). Construction 5.12 (Quasitoric manifold from characteristic pair). Given a point q 2 P n , we denoted by G(q) the minimal face containing q in its relative interior. Now set M 2n (`) := (T n P n )= ; where (t1 ; p) (t2 ; q) if and only if p = q and t1 t?1 2 `(G(q)) (compare with 2 Construction 5.5 for toric varieties). The free action of T n on T n P n obviously descends to an action on (T n P n )= , with quotient P n . The latter action is free over the interior of P n and has a xed point for each vertex of P n . Just as P n is covered by the open sets Uv , based on the vertices and di eomorphic to Rn (see + Construction 5.8), so the space (T n P n )= is covered by open sets (T n Uv )= homeomorphic to (T n Rn )= , and therefore to C n . This implies that the T n + action on (T n P n)= is locally standard, and therefore (T n P n)= is a quasitoric manifold. Definition 5.13. Given an automorphism : T n ! T n, say that two qua2 2 sitoric manifolds M1 n , M2 n over the same P n are -equivariantly di eomorphic 2 2 if there is a di eomorphism f : M1 n ! M2 n such that f (t x) = (t)f (x) for n , x 2 M 2n . The automorphism induces an automorphism all t 2 T of the 1 poset of subtori of T n. Any such automorphism descends to a -translation of characteristic pairs, in which the two characteristic maps di er by . The following proposition is proved as Proposition 1.8 in 48] (see also 38, Proposition 2.6]). Proposition 5.14. Construction 5.12 de nes a bijection between -equivariant di eomorphism classes of quasitoric manifolds and -translations of pairs (P n ; `). When is the identity, we deduce that two quasitoric manifolds are equivariantly di eomorphic if and only if their characteristic maps are the same.


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5.2.2. Cohomology of quasitoric manifolds. The cohomology ring structure of a quasitoric manifold is similar to that of a non-singular toric variety. To see this analogy we rst describe a cell decomposition of M 2n with even dimensional cells (a perfect cellular structure) and calculate the Betti numbers accordingly, following 48]. Construction 5.15. We recall the \Morse-theoretical arguments" from the proof of Dehn{Sommerville relations (Theorem 1.20). There we turned the 1skeleton of P n into a directed graph and de ned the index ind(v) of a vertex v 2 P n as the number of incident edges that point towards v. These inward edges span a b face Gv of dimension ind(v). Denote by Gv the subset of Gv obtained by deleting all b faces not containing v. Obviously, Gv is di eomorphic to Rind (v) and is contained in + n from Construction 5.8. Then ev := ?1 Gv is identi ed with b the open set Uv P ind (v) , and the union of the ev over all vertices of P n de ne a cellular decomposiC tion of M 2n . Note that all cells are even-dimensional and the closure of the cell ev is the facial submanifold M (Gv )2 ind(v) M 2n . This argument was earlier used by Khovanskii 84] for constructing perfect cellular decompositions of toric varieties. Proposition 5.16. The Betti numbers of M 2n vanish in odd dimensions, while in even dimensions are given by b2i (M 2n ) = hi (P n ); i = 0; 1; : : : ; n; where h(P n ) = (h0 ; : : : ; hn ) is the h-vector of P n . Proof. The 2i-th Betti number equals the number of 2i-dimensional cells in the cellular decomposition constructed above. This number equals the number of vertices of index i, which is hi (P n ) by the argument from the proof of Theorem 1.20.
Given a quasitoric manifold M 2n with characteristic map (5.4) and facet vectors i = ( 1i ; : : : ; ni )t 2 Zn, i = 1; : : : ; m, de ne linear forms (5.6) i := i1 v1 + + im vm 2 Z v1; : : : ; vm ]; 1 6 i 6 n: The images of these linear forms in the Stanley{Reisner ring Z(P n) will be denoted by the same letters. Lemma 5.17 (Davis and Januszkiewicz). 1 ; : : : ; n is a (degree-two) regular sequence in Z(P n). Let J` denote the ideal in Z(P n) generated by 1 ; : : : ; n . Theorem 5.18 (Davis and Januszkiewicz). The cohomology ring of M 2n is given by H (M 2n ; Z) = Z v1; : : : ; vm ]=(IP + J` ) = Z(P n)=J` ; where vi is the 2-dimensional cohomology class dual to the facial submanifold Mi2(n?1) (with arbitrary orientation chosen), i = 1; : : : ; m. We give proofs for the above two statements in section 6.5. Remark. Change of sign of vector i corresponds to passing from vi to ?vi in the description of the cohomology ring given by Theorem 5.18. We will use this observation in the next section.


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67

'' ' & &&
Quasitoric manifolds

5.2.3. Non-singular toric varieties and quasitoric manifolds. In this subsection we give a more detailed comparison of the two classes of manifolds. In general, none of these classes belongs to the other, and the intersection of the two classes contains smooth pro jective toric varieties as a proper subclass (see Figure 5.1). Below in this subsection we provide the corresponding examples.

Smooth pro jective toric varieties

Figure 5.1.

$ $ % %

Smooth complete toric varieties

$ %

Example 5.19. As it is suggested by comparing Constructions 5.5 and 5.12, a non-singular pro jective toric variety MP arising from a lattice simple polytope P n is a quasitoric manifold over the combinatorial type P n . The corresponding characteristic map ` : Fi 7! T (Fi ) is de ned by putting i = l i in (5.3). That is, the facet vectors are the normal vectors l i to facets of P n , i = 1; : : : ; m (see (1.1)). The corresponding characteristic n m-matrix is the matrix L from Construction 1.8. In particular, if P n is the standard simplex n (1.2) then MP is C P n (Example 5.7) and = (E j ? 1), where E is the unit n n-matrix and 1 is the column of units. See also Example 5.60 below. In general, a smooth non-pro jective toric variety may fail to be a quasitoric manifold: although the orbit space (for the T n-action) is a manifold with corners (see section 5.1), it may not be di eomorphic (or combinatorially equivalent) to a simple polytope. The authors are thankful to N. Strickland for drawing our attention to this fact. However, we do not know of any such example. Problem 5.20. Give an example of a non-singular toric variety which is not a quasitoric manifold. In 64, p. 71] one can nd an example of a complete non-singular fan in R3 which cannot be obtained by taking the cones with vertex 0 over the faces of a convex simplicial polytope. Nevertheless, since the corresponding simplicial complex K is a simplicial 2-sphere, it is combinatorially equivalent to a polytopal 2-sphere. This means that the corresponding non-singular toric variety M , although being non-pro jective, is still a quasitoric manifold. It is convenient to introduce the following notation here. Definition 5.21. We say that a simplicial fan in Rn is strongly polytopal (or simply polytopal ) if it can be obtained by taking the cones with vertex 0 over the faces of a convex simplicial polytope. Equivalently, a fan is strongly polytopal if it is a normal fan of simple lattice polytope (see Construction 5.4). Say that a simplicial fan is weakly polytopal if the underlying simplicial complex K is a polytopal sphere (that is, combinatorially equivalent to the boundary complex of a simplicial polytope).


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Suppose is a non-singular fan and M the corresponding toric variety. Then is strongly polytopal, and M is a quasitoric manifold if and only if is weakly polytopal. Thus, the answer to Problem 5.20 can be given by providing a non-singular fan which is not weakly polytopal. As it was told to the authors by Y. Civan (in private communications), this may be done by giving a (singular) fan whose underlying simplicial complex K is the Barnette sphere (see section 2.3) and then desingularizing it using the standard procedure (see 64, x2.6]). Combinatorial properties of the Barnette sphere obstruct the resulting (non-singular) fan to be weakly polytopal. On the other hand, it is easy to construct a quasitoric manifold which is not a toric variety. The simplest example is the manifold C P 2 # C P 2 , the connected sum of two copies of C P 2 . It is a quasitoric manifold over the square I 2 (this follows from the construction of equivariant connected sum, see 48, 1.11] or section 5.3 and Corollary 5.66 below). However, C P 2 # C P 2 do not admit even an almost complex structure (i.e., its tangent bundle cannot be made complex). The following problem arises. Problem 5.22. Let (P n ; `) be a characteristic pair (see De nition 5.11), and 2n (`) the derived quasitoric manifold (see Construction 5.12). Find conditions M on P n and ` so that M 2n (`) admits a T n-invariant complex (or almost complex) structure. The almost complex case of the above problem was formulated in 48, Problem 7.6]. Since every non-singular toric variety is a complex manifold, characteristic pairs coming from lattice simple polytopes (as described in Example 5.19) provide a su cient condition for Problem 5.22. However, this is not a necessary condition even for the existence of an invariant complex structure. Indeed, there exist smooth non-pro jective toric varieties coming from weakly polytopal fans (see the already mentioned example in 64, p. 71]). At the same time, we do not know any example of non-toric complex quasitoric manifold. Problem 5.23. Find an example of a non-toric quasitoric manifold that admits a T n -invariant complex structure. Although a general quasitoric manifold may fail to be complex or almost complex, it always admits a T n -invariant complex structure in the stable tangent bundle. The corresponding constructions are the sub ject of the next section. We will return to Problem 5.22 in subsection 5.4.2. Another class of problems arises in connection with the classi cation of quasitoric manifolds over a given combinatorial simple polytope. The general setting of this problem is discussed in section 5.5. Example 5.26 below shows that there are combinatorial simple polytopes that do not admit a characteristic map (and therefore cannot arise as orbit spaces for quasitoric manifolds). Problem 5.24. Give a combinatorial description of the class of polytopes P n that admit a characteristic map (5.4). A generalization of this problem is considered in chapter 7 (Problem 7.27). A characteristic map is determined by an integer n m-matrix which satises (5.5) for every vertex v 2 P n . The equation (det (v) )2 = 1 de nes a hypersurface in the space M(n; m; Z) of integer n m-matrices.

M is pro jective if and only if


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tersection (5.7)

Proposition 5.25. The set of characteristic matrices coincides with the in\

of hypersurfaces in the space M(n; m; Z), where v is running through the vertices of the polytope P n . Thus, Problem 5.24 is to determine for which polytopes the intersection in (5.7) is non-empty. Example 5.26 ( 48, Example 1.22]). Let P n be a 2-neighborly simple polytope with m > 2n facets (e.g., the polar of cyclic polytope C n (m) with n > 4 and m > 2n , see Example 1.17). Then this P n does not admit a characteristic map and therefore cannot appear as the quotient space of a quasitoric manifold. Indeed, by Proposition 5.25, it is su cient to show that intersection (5.7) is empty. Since m > 2n , any matrix 2 M(n; m; Z) (without zero columns) contains two columns, say i-th and j -th, which coincide modulo 2. Since P n is 2-neighborly, the corresponding facets Fi and Fj have non-empty intersection in P n . Hence, the columns i and j of enter the minor (v) for some vertex v 2 P n . This implies that the determinant of this minor is even and intersection (5.7) is empty. In particular, the above example implies that there are no non-singular toric varieties over the combinatorial polar cyclic polytope (C n (m)) with > 2n facets. This means that the combinatorial type C n (m) with m > 2n cannot be realized as a lattice simplicial polytope in such a way that the fan over its faces is nonsingular. In the toric geometry, the question of whether for any given complete simplicial fan there exists a combinatorially equivalent fan 0 that gives rise to a smooth toric variety was known as Ewald's conjecture of 1986. The rst counterexample was found in 67]. It was shown there that no fan over the faces of a lattice realization of C n (m) with m > n + 3 is non-singular. The comparison of this result with Example 5.26 suggests that some cyclic polytopes C n (m) with small number of vertices (between n + 3 and 2n ) may appear as the quotients of quasitoric manifolds, but not as quotients of non-singular toric varieties. Another interesting corollary of Example 5.26 is that the face ring Z(C n(m)) of (the boundary complex of ) C n (m) and the face ring Z=p(C n(m)) for any prime p does not admit a regular sequence of degree two (or a lsop). Of course, since Z=p(C n(m)) is Cohen{Macaulay, it admits a non-linear regular sequence. Note that in the case when k is of zero characteristic the ring k(C n (m)) always admits an lsop (and degree-two regular sequence) by Lemma 3.11.

v2P n

(det (v) )2 = 1

This section is the review of results obtained by N. Ray and the rst author in 37] and 38], supplied with some additional comments. A stably complex structure on a (smooth) manifold M is determined by a complex structure in the vector bundle (M ) Rk for some k, where (M ) is the tangent bund le of M and Rk denotes a trivial real k-dimensional bundle over M . A stably complex manifold (in other notations, weakly almost complex manifold or U -manifold ) is a manifold with xed stably complex structure, which one can view as a pair (M ; ), where is a complex bundle isomorphic, as a real bundle, to

5.3. Stably complex structures, and quasitoric representatives in cobordism classes


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(M ) Rk for some k. If M itself is a complex manifold, then it possesses the canonical stably complex structure (M ; (M )). The operations of disjoint union and product endow the set of cobordism classes M ; ] of stably complex manifolds with the structure of a graded ring, called the complex cobordism ring U . By the theorem of Milnor and Novikov, the complex cobordism ring is isomorphic to the polynomial ring on an in nite number of even-dimensional generators: U = Z a1; a2 ; : : : ]; deg ai = 2i; (see 104], 130]). The ring U is the coe cient ring for generalized (co)homology theory known as the complex (co)bordisms . We refer to 130] as the standard source for the cobordism theory. Stably complex manifolds was the main sub ject of F. Hirzebruch's talk at the 1958 International Congress of Mathematicians, see 135]. Using Milnor hypersurfaces (Example 5.39) and the Milnor{Novikov theorem it was shown by Milnor 135] that every complex cobordism class contains a non-singular algebraic variety, not necessarily connected. The following problem is still open. Problem 5.27 (Hirzebruch). Which complex cobordism classes in U contain connected non-singular algebraic varieties? A weaker version of this question, which is also open, asks which cobordism classes contain connected almost complex manifolds. Example 5.28. The 2-dimensional cobordism group U = Z is generated by 2 the class of C P 1 ] (Riemannian sphere). Every cobordism class k C P 1 ] 2 U 2 contains a non-singular algebraic variety, namely, the disjoint union of k copies of C P 1 for k > 0 and a Riemannian surface of genus (1 ? k) for k 6 0. However, connected algebraic varieties are contained only in the cobordism classes k C P 1 ] with k 6 1. The problem of choosing appropriate generators for the ring U is very important in the cobordism theory and its applications. As it was recently shown in 37] and 38], every complex cobordism class (of dimension > 2) contains a quasitoric manifold (see Theorem 5.38 below). By the de nition, quasitoric manifolds are necessarily connected, so the result may be considered as an answer to the quasitoric analogue of Hirzebruch's question. The construction of quasitoric representatives in complex cobordism classes relies upon an additional structure on a quasitoric manifold, called omniorientation , which provides a combinatorial description for canonical stably complex structures. Let : M 2n ! P n be a quasitoric manifold with characteristic map `. Since the torus T n (5.1) is oriented, a choice of orientation for P n is equivalent to a choice of orientation for M 2n . (An orientation of P n is speci ed by orienting the ambient space Rn .) Definition 5.29. An omniorientation of a quasitoric manifold M 2n consists of a choice of an orientation for M 2n and for every facial submanifold Mi2(n?1) = ?1 (Fi ), i = 1; : : : ; m. Thus, there are 2m+1 omniorientations in all for given M 2n . An omniorientation of M 2n determines an orientation for every normal bundle i := (Mi M 2n ), i = 1; : : : ; m. Since every i is a real 2-plane bundle, an orientation of i allows one to interpret it as a complex line bundle. The isotropy


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subgroup T (Fi ) (see (5.3)) of submanifold Mi2(n?1) = ?1 (Fi ) acts on the normal bundle i , i = 1; : : : ; m. Thus, we have the following statement. Proposition 5.30. A choice of omniorientation for M 2n is equivalent to a choice of orientation for P n together with an unambiguous choice of facet vectors i , i = 1; : : : ; m in (5.3). We refer to a characteristic map ` as directed if all circles `(Fi ), i = 1; : : : ; m, are oriented. This implies that the signs of facet vectors i = ( 1i ; : : : ; ni )t , i = 1; : : : ; m, are determined unambiguously. In the previous section we organized the facet vectors into the integer n m matrix . This matrix satis es (5.5). Due to (5.3), the matrix carries exactly the same information as a directed characteristic map. Let ZF denote the m-dimensional free Z-module spanned by the set F of facets of P n . Then de nes an epimorphism : ZF ! Zn by (Fi ) = i and an epimorphism T F ! T m, which we will denote by the same letter . In the sequel we write Zm for ZF and T m for T F , assuming that the member e i of the standard basis of Zm corresponds to the facet Fi 2 ZF, i = 1; : : : ; m, and similarly for T m. Definition 5.31. A directed characteristic pair (P n ; ) consists of a combinatorial simple polytope P n and an integer matrix (or, equivalently, an epimorphism : Zm ! Zn) that satis es (5.5). Proposition 5.30 shows that the characteristic pair of an omnioriented quasitoric manifold is directed. On the other hand, the quasitoric manifold derived from a directed characteristic pair using Construction 5.12 is omnioriented. Construction 5.32. The orientation of the normal bundle i over Mi denes an integral Thom class in the cohomology group H 2 (T ( i )), represented by a complex line bundle over the Thom complex T ( i ). We pull this back along the Pontryagin{Thom collapse M 2n ! T ( i ), and denote the resulting bundle i . The restriction of i to Mi M 2n is i . In the algebraic geometry this construction corresponds to assigning the line bundle to a divisor. In particular, in the case when M 2n is a smooth toric variety, the line bundle i corresponds to the divisor Di , see Theorem 5.3. Theorem 5.33 ( 48] and 38, Theorem 3.8]). Every omniorientation of a quasitoric manifold M 2n determines a stably complex structure on it by means of the fol lowing isomorphism of real 2m-bund les: (M 2n ) R2(m?n) = 1 m: The above isomorphism of real vector bundles is essentially due to Davis and Januszkiewicz (see 48, Theorem 6.6]). The interpretation of stably complex structures in terms of omniorientations was given in 38]. Corollary 5.34. In the notation of Theorem 5.18, suppose vi 2 H 2 (M 2n ) is the cohomology class dual to the oriented facial submanifold Mi of an omnioriented quasitoric manifold M 2n , i = 1; : : : ; m. Then the total Chern class of stably complex structure on M 2n de ned by the omniorientation is given by c(M 2n ) = (1 + v1 ) : : : (1 + vm ) 2 H (M 2n ): It follows from Theorem 5.33 that a directed characteristic pair (P n ; ) determines a complex cobordism class M 2n ; 1 m ] 2 U . The following direct


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extension of Theorem 5.18 provides a description of the complex cobordism ring of an omnioriented quasitoric manifold. Proposition 5.35 ( 38, Proposition 5.3]). Let vi denote the rst cobordism Chern class c1 ( i ) 2 2 (M 2n ) of the bund le i , 1 6 i 6 m. Then the complex U cobordism ring of M 2n is given by 2n U (M ) = U v1 ; : : : ; vm ]=(IP + J ); where the ideals IP and J are de ned in the same way as in Theorem 5.18. Note that the Chern class c1 ( i ) is Poincare dual to the inclusion Mi2(n?1) 2n by the construction of i . This highlights the remarkable fact that the complex M bordism groups U (M 2n ) are spanned by embedded submanifolds. By de nition, the fundamental cobordism class hM 2n i 2 2n (M 2n ) is dual to the bordism class of U a point. Thus, hM 2n i = vi1 vin for any set fi1 ; : : : ; in g such that Fi1 \ \ Fin is a vertex of P n . The following two examples are used to construct quasitoric representatives in complex cobordisms. Example 5.36 (bounded ag manifold 36]). A bounded ag in C n+1 is a complete ag U = fU1 U2 Un+1 = C n+1 g for which Uk , 2 6 k 6 n, contains k?1 spanned by the rst k ? 1 standard basis vectors. As the coordinate subspace C it is shown in 38, Example 2.8], the 2n-dimensional manifold Bn of all bounded ags in C n+1 is a quasitoric manifold over the combinatorial cube I n with respect to the action induced by t z = (t1 z1 ; : : : ; tn zn ; zn+1 ) on C n+1 , where t 2 T n. Example 5.37. A family of manifolds Bi;j (0 6 i 6 j ) is introduced in 37]. The manifold Bi;j consists of pairs (U; W ), where U is a bounded ag in C i+1 (see ? Example 5.36) and W is a line in U1 C j ?i . So Bi;j is a smooth C P j?1 -bundle over Bi . It is shown in 38, Example 2.9] that Bi;j is a quasitoric manifold over the j ?1 . product I i The canonical stably complex structures and omniorientations on the manifolds Bn and Bi;j are described in 38, examples 4.3, 4.5]. 2 2 Remark. The product of two quasitoric manifolds M1 n1 and M2 n2 over polyn1 and P n2 is a quasitoric manifold over P n1 P n2 . This construction topes P1 2 1 2 extends to omnioriented quasitoric manifolds and is compatible with stably complex structures (details can be found in 38, Proposition 4.7]). It is shown in 37] that the cobordism classes of Bi;j multiplicatively generate the ring U . Hence, every 2n-dimensional complex cobordism class may be represented by a disjoint union of products (5.8) Bi1 ;j1 Bi2 ;j2 Bik ;jk ; Pk where q=1 (iq + jq ) ? 2k = n. Each such component is a quasitoric manifold, under the product quasitoric structure. This result is the substance of 37]. The stably complex structures of products (5.8) are induced by omniorientations, and are therefore also preserved by the torus action. To give genuinely quasitoric representatives (which are, by de nition, connected) for each cobordism class of dimension > 2, it remains only to replace the disjoint union of products (5.8) with their connected sum. This is done in 38, x6] using Construction 1.13 and its extension to omnioriented quasitoric manifolds.


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Theorem 5.38 ( 38, Theorem 6.11]). In dimensions > 2, every complex cobordism class contains a quasitoric manifold, necessarily connected, whose stably complex structure is induced by an omniorientation, and is therefore compatible with the action of the torus. The connected sum operation usually destroys the algebraicity of manifolds, so the complex cobordism representatives provided by the above theorem in general are not algebraic (compare Example 5.28). We note that in their work 44, x42] Conner and Floyd constructed a class of manifolds with canonical circle actions which can be chosen as representatives for multiplicative generators in oriented cobordisms. One can show that these Conner{ Floyd manifolds can be obtained as particular cases of manifolds Bi;j from Example 5.37. However, Conner and Floyd did not consider actions of half-dimensional tori on their manifolds. Example 5.39. The standard set of multiplicative generators for U consists of pro jective spaces C P i , i > 0, and Milnor hypersurfaces Hi;j C P i C P j , 1 6 i 6 j . The hypersurface Hi;j is de ned by

Hi;j = (z0 :

n

: zi ) (w0 :

: wj ) 2 C P i C P j :

i X q=0

zq wq = 0 :

o

However, the hypersurfaces Hi;j are not quasitoric manifolds for i > 1, see 37]. This can be shown in the following way. Construction 5.40. Let C i+1 C j +1 be the subspace spanned by the rst i + 1 vectors of the standard basis of C j +1 . Identify C P i with the set of complex lines l C i+1 . To each line l assign the set of hyperplanes C j +1 that contain l. j ?1 , so there is a bundle E ! C P i with bre The latter set is identi ed with C P C P j?1 . Here E is the set of pairs (l; ), l , and the pro jection takes (l; ) to l. Lemma 5.41. Hi;j is identi ed with the total space of bund le E ! C P i . Proof. A line l C i+1 is given by a vector (z0 : z1 : : zi ). A hyperplane C j +1 is given by a linear form. If we denote the coe cients of this linear form by w0 ; w1 ; : : : ; wj , then the condition l is exactly that from the de nition of Hi;j . Theorem 5.42. The cohomology of Hi;j is given by
. H (Hi;j ) = Z u; v] ui+1 = 0; vj?i

i X k=0

uk vi?k = 0 ;

where deg u = deg v = 2. Proof. We will use the notation from Construction 5.40. Let denote the bundle over C P i whose bre over l 2 C P i is the j -dimensional subspace l? C j +1 . Then one can identify Hi;j with the pro jectivization C P ( ). Indeed, for any line l0 l? representing a point in the bre of C P ( ) over l 2 C P i the hyperplane = (l0 )? C j +1 contains l, so the pair (l; ) represents a point in Hi;j (see Lemma 5.41). The rest of the proof reproduces the general argument from the Dold theorem about the cohomology of pro jectivizations. Denote by the tautological line bundle over C P i (its bre over l 2 C P i is the line l itself ). Then is a trivial (j + 1)-dimensional bundle. Set w = c1 ( ) 2


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H 2 (C P i ). Let c( ) = 1 + c1 ( ) + c2 ( ) + : : : denote the total Chern class. Since c( )c( ) = 1 and c( ) = 1 ? w, we get (5.9) c( ) = 1 + w + + wi : Consider the pro jection p : C P ( ) ! C P i . Denote by the \tautological" line bundle over C P ( ) whose ber over a point l0 2 C P ( ) is the line l0 itself. Denote by ? the (j ? 1)-bundle over C P ( ) whose bre over a point l0 l? is the ? . Set orthogonal complement to l0 in l?. Then it is easy to see that p ( ) = 2 (C P ( )) and u = p (w) 2 H 2 (C P ( )). Then ui+1 = 0. We have v = c1 ( ) 2 H c( ) = 1 ? v and c(p ( )) = c( )c( ? ), hence, ? c( ? ) = p c( ) (1 ? v)?1 = (1 + u + + ui )(1 + v + v2 + : : : ) (see (5.9)). But ? is (j ? 1)-dimensional, hence cj ( ? ) = 0. Calculating the homogeneous part of degree j in the above identity, we get the second identity P 0 = vj?i i =0 uk vi?k . k Since both C P i and C P j?1 have only even-dimensional cells, the Leray{Serre spectral sequence of the bundle p : C P ( ) ! C P i collapses at the E2 term. It follows that there is an epimorphism Z u; v] ! H (C P ( )), and additively the cohomology of H (C P ( )) coincides with that of C P i C P j?1 . Hence, there are
no other relations except those two mentioned in the theorem. Proposition 5.43. Hi;j is not a quasitoric manifold for i > 1. Proof. By Theorem 5.18, the cohomology of a quasitoric manifold is isomorphic to a quotient Z v1; : : : ; vm ]=I +J , where the ideal I is generated by square-free monomials and J is generated by linear forms. Due to (5.5) we may assume without loss of generality that rst n variables v1 ; : : : ; vn are expressed via the last m ? n by means of linear equations with integer coe cients. Hence, we have Z v1; : : : ; vm ]=I +J = Z w1; : : : ; wm?n ]=I 0 ; where I 0 is an ideal having a basis each of whose elements is a product of > 2 integer linear forms. Suppose now that Hi;j , i > 1, is a quasitoric manifold. Then we have an isomorphism Z w1; : : : ; wm?n ]=I 0 = Z u; v]=I 00 ; where I 00 is the ideal from Theorem 5.42. It is easy to see that in this case we have m ? n = 2 above, and w1 ; w2 can be identi ed with u; v. Thus, the ideal I 00 must have a basis consisting of products of > 2 linear forms with integer coe cients. But this is impossible for i > 1.

The constructions from the previous section open the way to evaluation of cobordism invariants (Chern numbers, Hirzebruch genera etc.) on omnioriented quasitoric manifolds in terms of the combinatorics of the quotient. In this section we expose the results obtained in this direction by the second author in 111], 112]. Namely, using arguments similar to that from the proof of Theorem 1.20 we construct a circle action with only isolated xed points on any quasitoric manifold M 2n . If M 2n is omnioriented then this action preserves the stably complex structure and

5.4. Combinatorial formulae for Hirzebruch genera of quasitoric manifolds


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75

its local representations near xed points are described in terms of the characteristic matrix . This allows us to calculate Hirzebruch's y -genus as a sum of contributions corresponding to the vertices of polytope. Each of these contributions depends only on the \local combinatorics" near the vertex. In particular, we obtain formulae for the signature and the Todd genus of M 2n . Definition 5.44. The Hirzebruch genus 74], 75] associated with the series X Q(x) = 1 + qk xk ; qk 2 Q ; is the ring homomorphism 'Q : U ! Q that to each cobordism class M 2n ] 2 Un 2 assigns the value given by the formula

'Q M ] =

2n

n Y

Here M 2n is a smooth manifold whose stable tangent bundle (M 2n ) is a complex bundle with complete Chern class in cohomology

i=1

Q(xi ); hM 2n i :
n Y

c( ) = 1 + c1 ( ) +

+ cn ( ) =

and hM 2n i is the fundamental class in homology. As it was shown by Hirzebruch, every ring homomorphism ' : U ! Q arises as a genus 'Q for some Q. There is also an oriented version of Hirzebruch genera, which deals with ring homomorphisms ' : SO ! Q from the oriented cobordism ring SO . Definition 5.45. The y -genus is the Hirzebruch genus associated with the series + ye?x(1+y) Q(x) = x(1 ? e?x(1+y) ) ; 1 where y 2 R is a parameter. For particular values y = ?1; 0; 1 we obtain the n-th Chern number, the Todd genus and the L-genus of M 2n correspondingly. Given a 4k-dimensional oriented manifold X 4k , the signature sign(X 4k ) is dened as the signature (the number of positive squares minus the number of negative ones) of the intersection form ? f ( ; ) := ; hX 4k i ; ; 2 H 2k (X 4k ) in the middle-dimensional cohomology H 2k (X 4k ). We also extend the signature to all even-dimensional manifolds by setting sign(X 4k+2 ) = 0. It can be shown that the signature is multiplicative and is an invariant of cobordism, so it de nes a ring homomorphism ' : SO ! Z (a genus). By the classical theorem of Hirzebruch 74], the signature coincides with the L-genus, and we will not distinguish between the two notions in the sequel. In the case when M 2n is a complex manifold, the value y (M 2n ) can be calculated in terms of Euler characteristics of the Dolbeault complexes on M 2n , see 74]. This was the original Hirzebruch's motivation for studying the y -genus. In this section we assume that we are given an omnioriented quasitoric manifold M 2n over some P n with characteristic matrix . This speci es a stably complex structure on M 2n , as described in the previous section. The orientation of M 2n determines the fundamental class hM 2n i 2 H2n (M 2n ; Z).

i=1

(1 + xi );


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5. TORIC AND QUASITORIC MANIFOLDS

action and calculate the y -genus. Construction 5.46. Suppose v is a vertex of P n expressed as the intersection of n facets: (5.10) v = Fi1 \ \ Fin : To each facet Fik above assign the unique edge Ek such that Ek \ Fik = v (that T is, Ek = j6=k Fij ). Let e k be a vector along Ek with origin v. Then e 1 ; : : : ; e n is a basis of Rn , which may be either positively or negatively oriented depending on the ordering of facets in (5.10). Throughout this section we assume this ordering to be such that e 1 ; : : : ; e n is a positively oriented basis. Once we speci ed an ordering of facets in (5.10), the facet vectors i1 ; : : : ; in at v may in turn constitute either positively or negatively oriented basis depending on the sign of the determinant of (v) = ( i1 ; : : : ; in ) (see (5.5)). Definition 5.47. The sign of a vertex v = Fi1 \ \ Fin is (v) := det (v) : One can understand the sign of a vertex geometrically as follows. For each vertex v 2 P n the omniorientation of M 2n determines two orientations of the tangent space Tv M 2n at v. The rst is induced by the orientation of M 2n . On the other hand, Tv M 2n decomposes into the sum of n two-dimensional vector spaces normal to the facial submanifolds Mi1 ; : : : ; Min containing v. By the de nition of omniorientation, each of these two-dimensional vector spaces is oriented, so they together de ne another orientation of Tv M 2n . Then (v) = 1 if the two orientations coincide and (v) = ?1 otherwise. The collection of signs of vertices of P n is an important invariant of an omnioriented quasitoric manifold. Note that reversing the orientation of M 2n changes all signs (v) to the opposite. At the same time changing the direction of one facet vector reverses the signs for those vertices contained in the corresponding facet. Let E be an edge of P n . The isotropy subgroup of 2-dimensional submanifold ?1 (E ) M 2n is an (n ? 1)-dimensional subtorus, which we denote by T (E ). It can be written as ? (5.11) T (E ) = e2 i'1 ; : : : ; e2 i'n 2 T n : 1 '1 + : : : + n 'n = 0 for some integers 1 ; : : : ; n . We refer to := ( 1 ; : : : ; n )t as the edge vector corresponding to E . This is a primitive vector in the dual lattice (Zn) and is determined by E only up to a sign. There is no canonical way to choose these signs simultaneously for all edges. However, the following lemma shows that the omniorientation of M 2n provides a canonical way to choose signs of edge vectors \locally" at each vertex. Lemma 5.48. For each vertex v 2 P n , the signs of edge vectors 1 ; : : : ; n meeting at v can be chosen in such a way that the n n-matrix M(v) := ( 1 ; : : : ; n ) satis es the identity Mtv) (v) = E; ( where E is the unit matrix. In other words, 1 ; : : : ; n and i1 ; : : : ; in are conjugate bases.

5.4.1. The sign and index of a vertex, edge vectors and calculation of the y -genus. Here we introduce some combinatorial invariants of the torus


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77

Proof. At the beginning we choose signs of the edge vectors at v arbitrary, and express v as in (5.10). Then k is the edge vector corresponding to the edge Ek opposite to Fik , k = 1; : : : ; n. It follows that Ek Fil for l 6= k, so T (Fil ) T (Ek ). Hence, (5.12) h k ; il i = 0; l 6= k; (see (5.3) and (5.11)). Since k is a primitive vector and i1 ; : : : ; in is a basis of Zn, it follows from (5.12) that h k ; ik i = 1. Changing the sign of ik if necessary, we obtain h k ; ik i = 1; which together with (5.12) gives Mtv) (v) = E, as needed. ( In the sequel, while making some local calculations near a vertex v, we assume that the signs of edge vectors are chosen as in the above lemma. It follows that the edge vectors 1 ; : : : ; n meeting at v constitute an integer basis of Zn and det M(v) = (v): (5.13) Example 5.49. Suppose M 2n = MP is a smooth toric variety arising from a lattice simple polytope P de ned by (1.1). Then i = l i , i = 1; : : : ; m (see Example 5.19), whereas the edge vectors at v 2 P n are the primitive integer vectors e 1 ; : : : ; e n along the edges with origin at v . It follows from Construction 5.46 that (v) = 1 for any v (compare with Proposition 5.53 below). Lemma 5.48 in this case expresses the fact that e 1 ; : : : ; e n and l i1 ; : : : ; l in are conjugate bases of Zn. Remark. Globally Lemma 5.48 provides two directions (signs) for an edge vector, one for each of its ends. These two signs are always di erent if M 2n is a complex manifold (e.g. a smooth toric variety), but in general they may be the same as well. Let = ( 1 ; : : : ; n )t 2 Zn be a primitive vector such that (5.14) h ; i 6= 0 for any edge vector : The vector de nes a one-dimensional oriented subtorus: ? T := e2 i 1 ' ; : : : ; e2 i n ' 2 T n ; ' 2 R : Lemma 5.50 ( 112, Theorem 2.1]). For any satisfying (5.14) the circle T acts on M 2n with only isolated xed points, corresponding to the vertices of P n . For each vertex v = Fi1 \ \ Fin the action of T induces a representation of S 1 in the tangent space Tv M 2n with weights h 1 ; i; : : : ; h n ; i. Remark. If M 2n = MP is a smooth toric variety, then the genericity condition (5.14) is equivalent to that from the proof of Theorem 1.20. Definition 5.51. Suppose we are given a primitive vector satisfying (5.14). De ne the index of a vertex v 2 P n as the number of negative weights of the S 1 -representation in Tv M 2n from Lemma 5.50. That is, if v = Fi1 \ \ Fin then ind (v) = f#k : h k ; i < 0g: Remark. The index of a vertex v can be also de ned in terms of the facet vectors at v. Indeed, Lemma 5.48 shows that if v = Fi1 \ \ Fin then = h 1 ; i i1 + + h n ; i in :


78

5. TORIC AND QUASITORIC MANIFOLDS

Hence, ind (v) equals the number of negative coe cients in the representation of as a linear combination of basis vectors i1 ; : : : ; in . Theorem 5.52 ( 111, Theorem 6], 112, Theorem 3.1]). For any vector satisfying (5.14), the y -genus of M 2n can be calculated as X (?y)ind (v) (v): y (M 2n ) = The proof of this theorem uses the Atiyah{Hirzebruch formula 8] and the circle action from Lemma 5.50. 5.4.2. Top Chern number and Euler characteristic. The value of the y -genus y (M 2n ) at y = ?1 equals the n-th Chern number cn ( )hM 2n i for any 2n-dimensional stably complex manifold M 2n ; ]. If the stably complex structure on M 2n comes from a complex structure in the tangent bundle (i.e. if M 2n is almost complex), then the n-th Chern number equals the Euler characteristic of M 2n . However, for general stably complex manifolds, the two numbers may di er, see Example 5.61 below. Given an omnioriented quasitoric manifold M 2n , Theorem 5.52 gives the following formula for its top Chern number: X (5.15) cn M 2n ] = (v): If M is a smooth pro jective toric variety, then (v) = 1 for every vertex v 2 P n (see Example 5.49) and cn M 2n ] equals the Euler characteristic e(M 2n ). Hence, for toric varieties the Euler characteristic equals the number of vertices of P n , which of course is well known. This is also true for arbitrary quasitoric M 2n : (5.16) e(M 2n ) = fn?1(P n ): To prove this, one can just use Lemma 5.50 and observe that the Euler characteristic of an S 1 -manifold equals the sum of Euler characteristics of xed submanifolds. Comparing (5.15) and (5.16), we can deduce some results on the existence of a T n-invariant almost complex structure on a quasitoric manifold M 2n (see Problem 5.22). An almost complex structure on M 2n determines a canonical orientation of the manifold. A T n-invariant almost complex structure also determines orientations for the facial submanifolds Mi2(n?1) M 2n , i = 1; : : : ; m (since they are xed point sets for the appropriate subtori) and thus gives rise to an omniorientation of M 2n . Proposition 5.53. Suppose that an omniorientation of a quasitoric manifold M 2n is determined by a T n-invariant almost complex structure. Then (v) = 1 for any vertex v 2 P n and, therefore, cn M 2n ] = e(M 2n ): Proof. Indeed, the tangent space Tv M 2n has canonical complex structure, and the orientations of normal subspaces to facial submanifolds meeting at v are the canonical orientations of complex subspaces. Hence, the two orientations of Tv M 2n coincide, and (v) = 1. As a corollary, we obtain the following necessary condition for the existence of a T n-invariant almost complex structure on M 2n .
2n

v2P n

v2P n


5.4. HIRZEBRUCH GENERA OF QUASITORIC MANIFOLDS

79

the corresponding characteristic matrix (with undetermined signs of column vectors). Suppose M 2n admits a T n -invariant almost complex structure. Then the signs of column vectors of can be chosen in such a way that the minors (v) (see (5.5)) are positive for al l vertices v = Fi1 \ \ Fin of P n . On the other hand, due to a theorem of Thomas 132, Theorem 1.7], a real orientable 2n-bundle has a complex structure if and only if it has a stable complex structure ! such that cn (!) = e( ) (the latter denotes the Euler class). It follows from (5.15) and (5.16) that the condition from the above corollary is also su cient for a quasitoric manifold M 2n to admit an almost complex structure (not necessarily T n-invariant). Note that although the stably complex structure determined by an omniorientation of a quasitoric manifold is T n -invariant (see Theorem 5.38), the almost complex structure whose existence is claimed by the result of Thomas (provided that the condition cn M 2n ] = e(M 2n) is satis ed) may fail to be invariant.

Corollary 5.54. Let M 2n be a quasitoric manifold over P n and

L-genus). Theorem 5.52 gives the following formula.
can be calculated as

5.4.3. Signature. The value of the y -genus at y = 1 is the signature (or the
Corollary 5.55. The signature of an omnioriented quasitoric manifold M 2n

sign(M 2n ) =

X

Being an invariant of an oriented cobordism class, the signature does not depend on a particular choice of stably complex structure (or omniorientation) on the oriented manifold M 2n . The following modi cation of Corollary 5.55 provides a formula for sign(M 2n ) that does not depend on an omniorientation. Corollary 5.56 ( 112, Corollary 3.3]). The signature of an oriented quasitoric manifold M 2n can be calculated as X sign(M 2n ) = det( e 1 ; : : : ; e n );
where ek , k h ek ; i > 0. If M 2n = Corollary 5.55

v2P n

(?1)

ind (v)

(v):

= 1; : : : ; n, are the edge vectors at v oriented in such a way that

v2P n

MP is a smooth toric variety, then (v) = 1 for any v 2 P n , and
gives sign(MP ) =
X

Since in this case ind (v) equals the index from the proof of Theorem 1.20, we obtain (5.17) sign(MP ) =
n X

v2P n

(?1)ind (v) :

Note that if n is odd then the right hand side of to the Dehn{Sommerville equations. The formula context in recent work of Leung and Reiner 90]. side of (5.17) arises in the following combinatorial

k=1

(?1)k hk (P ): the above formula vanishes due (5.17) appears in a more general The quantity in the right hand conjecture.


80

5. TORIC AND QUASITORIC MANIFOLDS

Problem 5.57 (Charney{Davis conjecture). Let K be a (2q ? 1)-dimensional Gorenstein* ag complex with h-vector (h0 ; h1 ; : : : ; h2q ). Is it true that (?1)q (h0 ? h1 + + h2q ) > 0? This conjecture was posed in 41, Conjecture D] for ag simplicial homology spheres. Stanley 129, Problem 4] extended it to Gorenstein* complexes. The Charney{ Davis conjecture is closely connected with the following di erential-geometrical conjecture. Problem 5.58 (Hopf conjecture). Let M 2q be a Riemannian manifold of nonpositive sectional curvature. Is it true that the Euler characteristic (M 2n ) satis es the inequality (?1)q (M 2q ) > 0? More details about the connection between the Charney{Davis and Hopf conjectures can be found in 41] and in a more recent paper 50]. The relationships between the above two problems and the signature of a toric variety are discussed in 90].

5.4.4. Todd genus. The next important particular case of y -genus is the Todd genus , corresponding to y = 0. In this case the summands in the formula from Theorem 5.52 are not well de ned for the vertices of index 0, so it requires some additional analysis. Theorem 5.59 ( 111, Theorem 7], 112, Theorem 3.4]). The Todd genus of an omnioriented quasitoric manifold can be calculated as X td(M 2n ) = (v)
v2P n : ind (v)=0

(the sum is taken over al l vertices of index 0). In the case of smooth toric variety there is only one vertex of index 0. This is the \bottom" vertex of P n , which has all incident edges pointing out (in the notations used in the proof of Theorem 1.20). Since (v) = 1 for every v 2 P n , Theorem 5.59 gives td(MP ) = 1, which is well known (see e.g. 64, x5.3]). Note that for algebraic varieties the Todd genus equals the arithmetic genus 74]. If M 2n is an almost complex manifold then td(M 2n ) > 0 by Proposition 5.53 and Theorem 5.59.

5.4.5. Examples.

variety. Its stably complex structure is determined by the standard complex structure in C P 2 , that is, via the isomorphism of bundles (C P 2 ) C ' . Here C is the trivial complex line bundle and is the Hopf line bundle over C P 2 . The orientation is de ned by the complex structure. The toric variety C P 2 arises from the 2-dimensional lattice simplex with vertices (0; 0), (1; 0) and (0; 1). The facet vectors here are primitive, normal to facets, and pointing inside the polytope. The edge vectors are primitive, parallel to edges, and pointing out of the corresponding vertex. This is shown in Figure 5.2. Let us calculate the Todd genus and the signature using Corollary 5.55 and Theorem 5.59. We have (v1 ) = (v2 ) = (v3 ) = 1.

Example 5.60. Let us look at the pro jective space C P 2 regarded as a toric


5.4. HIRZEBRUCH GENERA OF QUASITORIC MANIFOLDS

81

Take = (1; 2), then ind(v1 ) = 0, ind(v2 ) = 1, ind(v3 ) = 2 (remember that the index is the number of negative scalar products of edge vectors with ). Thus, sign(C P 2 ) = sign(C P 2 ; ) = 1; td(C P 2 ) = td(C P 2 ; ) = 1:
v3
(0

; ?1)

@

@

(1

; ?1)
@ @

@

1

= (1; 0)

(0 1)

;

v1

(1 0)

;

"!
@ @

@

3

=(

?1; ?
@ @

1)

@ @

@

( 1 1)

?;

( 1 0)

2

= (0; 1)

?;

v2

Figure 5.2. (C P 2 )

C'

Taking = (1; 2), we nd ind (v1 ) = 0, ind (v2 ) = 0, ind (v3 ) = 1. Thus, sign C P 2 ; ] = 1; td C P 2 ; ] = 0: Note that in this case formula (5.15) gives cn C P 2 ; ] = (v1 ) + (v2 ) + (v3 ) = ?1; 2 is 3. while the Euler number of C P T n-equivariant stably complex and almost complex manifolds were considered in works of Hattori 71] and Masuda 93] as a separate generalization (called the unitary toric manifolds ) of toric varieties. Instead of Davis and Januszkiewicz's characteristic maps, Masuda in 93] used the notion of multi-fan to describe the combinatorial structure of the orbit space. The multi-fan is a collection of cones which may overlap unlike a usual fan. The Todd genus of a unitary toric manifold was calculated in 93] via the degree of the overlap of cones in the multi-fan. This result is equivalent to our Theorem 5.59 in the case of quasitoric manifolds. A formula for the y -genus similar to that from Theorem 5.52 has been obtained (independently) in a more recent paper 73]. For more information about multifans see 72].

the three facet vectors 1 ; 2 ; 3 , shown in Figure 5.3. This omniorientation di ers from the previous example by the sign of 3 . The corresponding stably complex structure is determined by the isomorphism (C P 2 ) R2 = . Using (5.13) we calculate 10 1 (v1 ) = 0 1 = 1; (v2 ) = ?1 1 = ?1; (v3 ) = 0 ?1 = ?1: 10 1

Example 5.61. Now consider C P 2 with the omniorientation determined by


82

5. TORIC AND QUASITORIC MANIFOLDS

v3
(0 1)

;

@

@

(1

;?

1)

@

@ @

1

= (1; 0)

(0 1)

;

v1

(1 0)

;

"!
@ @

@

3

= (1; 1)

@ @

@

(1 0)

;

@ @

( 1 1)

?;

2

= (0; 1)

v2

Figure 5.3. (C P 2 )

C'

There are two main classi cation problems for quasitoric manifolds over a given simple polytope: the equivariant (i.e. up to a -equivariant di eomorphism) and the topological (i.e. up to a di eomorphism). Due to Proposition 5.14, the equivariant classi cation reduces to describing all characteristic maps for given simple polytope P n . The topological classi cation problem usually requires additional analysis. In general, both problems seem to be intractable. However, in some particular cases nice classi cation results may be achieved. Here we give a brief review of what is known on the sub ject. Let M 2n be a quasitoric manifold over P n with characteristic map `. We assume here that the facets are ordered in such a way that the rst n of them share a common vertex. Lemma 5.62. Up to -equivalence (see De nition 5.13) , we may assume that `(Fi ) is the i-th coordinate subtorus Ti T n, i = 1; : : : ; n. Proof. Since the one-dimensional subtori `(Fi ), i = 1; : : : ; n, generate T n, we may de ne as any automorphism of T n that maps `(Fi ) to Ti . It follows that M 2n admits an omniorientation whose corresponding characteristic n m-matrix has the form (E j ), where E is the unit matrix and denotes some integer n (m ? n)-matrix. In the simplest case P n = n the equivariant (and topological) classi cation of quasitoric manifolds reduces to the following easy result. Proposition 5.63. Any quasitoric manifold over n is -equivariantly diffeomorphic to C P n (regarded as a toric variety, see Examples 5.7 and 5.19). Proof. The characteristic map for C P n has the form `CP n (Fi ) = Ti ; i = 1; : : : ; n; `CP n (Fn+1 ) = Sd ; 1 := f(e2 i' ; : : : ; e2 i' ) 2 T n g, ' 2 R, is the diagonal subgroup in T n . Let where Sd M 2n be a quasitoric manifold over n with characteristic map `M . We may assume

5.5. Classi cation problems


5.5. CLASSIFICATION PROBLEMS

83

that `M (Fi ) = Ti , i = 1; : : : ; n, by Lemma 5.62. Then it easily follows from (5.5) that ? `M (Fn+1 ) = e2 i"1 ' ; : : : ; e2 i"n ' 2 T n ; ' 2 R; where "i = 1, i = 1; : : : ; n. Now de ne the automorphism : T n ! T n by ? 2 i' ? e 1 ; : : : ; e2 i'n = e2 i"1 '1 ; : : : ; e2 i"n 'n : It can be readily seen that `M = `C P n , which together with Proposition 5.14 completes the proof. Both problems of equivariant and topological classi cation also admit a complete solution for n = 2 (i.e., for quasitoric manifolds over polygons). Example 5.64. Given an integer k, the Hirzebruch surface Hk is the complex manifold C P ( k C ), where k is the complex line bundle over C P 1 with rst Chern class k, and C P ( ) denotes the pro jectivisation of a complex bundle. In particular, each Hirzebruch surface is the total space of the bundle Hk ! C P 1 with bre C P 1 . The surface Hk is di eomorphic to S 2 S 2 for even k and to C P 2 # C P 2 for odd k, where C P 2 denotes the space C P 2 with reversed orientation. Each Hirzebruch surface is a non-singular pro jective toric variety, see 64, p. 8]. The orbit space for Hk (regarded as a quasitoric manifold) is a combinatorial square; the corresponding characteristic maps can be described using Example 5.19 (see also 48, Example 1.19]). Theorem 5.65 ( 106, p. 553]). A quasitoric manifold of dimension 4 is equivariantly di eomorphic to an equivariant connected sum of several copies of C P 2 and Hirzebruch surfaces Hk . Corollary 5.66. A quasitoric manifold of dimension 4 is di eomorphic to a connected sum of several copies of C P 2 , C P 2 and S 2 S 2 . The classi cation problem for quasitoric manifolds over a given simple polytope can be considered as a generalization of the corresponding problem for non-singular toric varieties. The classi cation result for 4-dimensional toric varieties is similar to Theorem 5.65 and can be found e.g., in 62]. In 105], to every toric variety over a simple 3-polytope P 3 there were assigned two integer weights on every edge of the dual simplicial complex KP . Using the special \monodromy conditions" for weights, the complete classi cation of toric varieties over simple 3-polytopes with 6 8 facets was obtained in 105]. A similar construction was used in 88] to obtain the classi cation of toric varieties over P n with m = n + 2 facets (note that any such simple polytope is a product of two simplices). In 56] the construction of weights from 105] was generalized to the case of quasitoric manifolds. This resulted in a criterion 56, Theorem 3] for the existence of a quasitoric manifold with prescribed weight set and signs of vertices (see Definition 5.47). The methods of 56] allow one to simplify the equations (5.5) for characteristic map on a given polytope. As an application, results on the classi cation of quasitoric manifolds over a product of an arbitrary number of simplices were obtained there.


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5. TORIC AND QUASITORIC MANIFOLDS


CHAPTER 6

Moment-angle complexes
For any combinatorial simple polytope P n with m facets, Davis and Januszkiewicz introduced in 48] a T m-manifold ZP with orbit space P n . This manifold has the following universal property: for every quasitoric manifold : M 2n ! P n there is a principal T m?n-bundle ZP ! M 2n whose composite map with is the orbit map for ZP . Topology of manifolds ZP and their further generalizations is very nice itself and at the same time provides an e ective tool for understanding inter-relations between algebraic and combinatorial ob jects such as Stanley{Reisner rings, subspace arrangements, cubical complexes etc. In this section we reproduce the original de nition of ZP and adjust it in a way convenient for subsequent generalizations. Let F = fF1 ; : : : ; Fm g be the set of facets of P n . For each facet Fi 2 F denote by TFi the one-dimensional coordinate subgroup of T F = T m corresponding to Fi . Then assign to every face G the coordinate subtorus Y TG = TFi T F : Note that dim TG = codim G. Recall that for every point q 2 P n we denoted by G(q) the unique face containing q in the relative interior. Definition 6.1. For any combinatorial simple polytope P n introduce the identi cation space ZP = (T F P n )= ; where (t1 ; p) (t2 ; q) if and only if p = q and t1 t?1 2 TG(q) . 2 Remark. The above de nition resembles constructions 5.5 and 5.12, but this time the equivalence relation depends only on the combinatorics of P n . Similar constructions appeared in earlier works of Vinberg 137] and Davis 47] on re ection groups. The free action of T m on T F P n descends to an action on ZP , with quotient P n . Let : ZP ! P n be the orbit map. The action of T m on ZP is free over the interior of P n , while each vertex v 2 P n represents the orbit ?1 (v) with maximal isotropy subgroup of dimension n. Lemma 6.2. The space ZP is a smooth manifold of dimension m + n. We will provide several di erent proofs of this lemma, each of which arises from an equivalent de nition of ZP . To give our rst proof we need the following simple topological fact. Proposition 6.3. The torus T k admits an embedding into Rk+1 .
85

6.1. Moment-angle manifolds ZP de ned by simple polytopes

Fi G


86

6. MOMENT-ANGLE COMPLEXES

Proof. The statement is obvious for k = 1. Suppose it holds for k = i ? 1. We may assume that T i?1 is embedded into an i-ball Di Ri . Represent the (i + 1)-sphere as S i+1 = Di S 1 S i?1 D2 (two pieces are glued by the identity di eomorphism of the boundaries). By the assumption, the torus T i = T i?1 S 1 can be embedded into Di S 1 and therefore into S i+1 . Since T i is compact and S i+1 is the one-point compacti cation of Ri+1 we have T i Ri+1 , and the statement follows by induction. Proof of Lemma 6.2. Construction 5.8 provides the atlas fUv g for P n as a manifold with corners. The set Uv is based on the vertex v and is di eomorphic to Rn . Then ?1 (Uv ) = T m?n R2n . We claim that T m?n R2n can be realized + as an open set in Rm+n , thus providing a chart for ZP . To see this we embed T m?n into Rm?n+1 as a closed hypersurface H (Proposition 6.3). Since the normal bundle is trivial, a small neighborhood of H Rm?n+1 is homeomorphic to T m?n R. Taking the cartesian product with R2n?1 we obtain an open set in Rm+n homeomorphic to T m?n R2n . The following statement follows easily from the de nition of ZP . Proposition 6.4. If P = P1 P2 for some simple polytopes P1 , P2 , then ZP = ZP1 ZP2 . If G P is a face, then ZG is a submanifold of ZP . Suppose now that we are given a characteristic map ` on P n and M 2n (`) is the derived quasitoric manifold (Construction 5.12). Choosing an omniorientation in any way we obtain a directed characteristic map : T F ! T n. Denote its kernel by H (`) (it depends only on `); then H (`) is an (m ? n)-dimensional subtorus of T F . Proposition 6.5. The subtorus H (`) acts freely on ZP , thereby de ning a principal T m?n-bund le ZP ! M 2n (`). Proof. It follows from (5.5) that H (`) meets every isotropy subgroup only at the unit. This implies that the action of H (`) on ZP is free. By de nitions of ZP and M 2n (`), the pro jection id : T F P n ! T n P n descends to the pro jection (T F P n )= ?! (T n P n )= ; which displays ZP as a principal T m?n-bundle over M 2n (`). To simplify notations, from now on we will write T m, C m etc. instead of T F , F etc. C Consider the unit poly-disc (D2 )m in the complex space: (D2 )m = (z1 ; : : : ; zm) 2 C m : jzi j 6 1; i = 1; : : : ; m : Then (D2 )m is stable under the standard action of T m on C m , and the quotient is the unit cube I m Rm . + Lemma 6.6. The cubical embedding iP : P n ! I m from Construction 4.5 is covered by an equivariant embedding ie : ZP ! (D2 )m . n Proof. Recall that the cubical complex C (P n ) consists of the cubes Cv based n . Note that C n is contained in the open set Uv P n (see on the vertices v 2 P nv Construction 5.8). The inclusion Cv Uv is covered by an equivariant inclusion n Bv C m , where Bv = ?1 (Cv ) is a closed subset homeomorphic to (D2 )n m?n . Since ZP = S n Bv and Bv is stable under the T m -action, the resulting T v2P embedding ZP ! (D2 )m is equivariant.


6.2. GENERAL MOMENT-ANGLE COMPLEXES ZK

87

then

P fn?1(P ) closed T m-invariant subspaces Bv . In section 6.3 we will use this to construct a cell decomposition of ZP . For now, we mention that if v = Fi1 \ \ Fin

It follows from the proof that the manifold Z is represented as a union of

or, more precisely, ie (Bv ) = (z1 ; : : : ; zm) 2 (D2 )m : jzi j = 1 for i 2 fi1; : : : ; in g : = n correspond to the maximal simplices in the polyRecalling that the vertices of P topal sphere KP (the boundary of polar polytope P ), we can write (6.1) (D2 ) Tb (D2 )m ; ie (ZP ) = where b = m] n . The above formula may be regarded as an alternative de nition of ZP . Introducing the polar coordinates in (D2 )m we see that ie (Bv ) is parametrized by n radial (or moment) and m angle coordinates. We refer to ZP as the moment-angle manifold corresponding to P n . Example 6.7. Let P n = n (the n-simplex). Then ZP is homeomorphic to the (2n + 1)-sphere S 2n+1 . The cubical complex C ( n ) (see Construction 4.5) n n consists of (n+1) cubes Cv . Each subset Bv = ?1 (Cv ) is homeomorphic to (D2 )n 1 . In particular, for n = 1 we obtain the representation of the 3-sphere S 3 as a S union of two solid tori D2 S 1 and S 1 D2 , glued by the identity di eomorphism of their boundaries. Another way to construct an equivariant embedding of ZP into C m can be derived from Construction 1.8. Construction 6.8. Consider the a ne embedding AP : P n ,! Rm de ned + by (1.6). It is easy to see that ZP enters the following pullback diagram:
2KP

ie (Bv ) = (D2 )n1 ;::: ;i i

n

T m?ni1 ;::: ;in g (D2 )m ; m]nf

ZP ? ?? C ?!
? ? y

? ? y

m

Thus, there is an equivariant embedding ZP ,! C m covering AP . A choice of matrix W in Construction 1.8 gives a basis in the (m ? n)-dimensional subspace orthogonal to the n-plane containing AP (P n ) (see (1.6)). The following statement follows. Corollary 6.9 (see also 38, x3]). The embedding ZP ,! C m has the trivial normal bund le. In particular, ZP is nul l-cobordant. Remark. Another way to see that ZP is null-cobordant is to establish a free S 1 -action on it (see e.g. Proposition 7.29). Then we get the manifold ZP S1 D2 with boundary ZP .
K m , we de ne a certain T m -stable In this section, for any cubical subcomplex in I subcomplex in the m-disc (D2 )m . In particular, this provides an extension of the construction of ZP to the case of general simplicial complex K . The resulting space,

P

n ? AP ! Rm ???
+

:

6.2. General moment-angle complexes Z


88

6. MOMENT-ANGLE COMPLEXES

denoted ZK , is not a manifold for arbitrary K , but is so when K is a simplicial sphere. The complex ZK , as a generalization of manifold ZP , rst appeared in 48, x4.1]. The approach used there involves the notion of \simple polyhedral complex", which extends the correspondence between polytopal simplicial spheres and simple polytopes to general simplicial complexes. In the sequel, we denote the canonical pro jection (D2 )m ! I m (and any of its restriction to a closed T m-stable subset of (D2 )m ) by . For each face C of I m (see (4.1)) de ne (6.2) B := ?1 (C ) = f(z1; : : : ; zm) 2 (D2 )m : zi = 0 for i 2 ; jzi j = 1 for i 2 g: = It follows that if j j = i and j j = j , then B = (D2 )j?i T m?j , where the disc factors D2 (D2 )j?i are indexed by n , while the circle factors S 1 T m?j are indexed by m] n . Definition 6.10. Let C be a cubical subcomplex of I m . The moment-angle complex ma(C ) corresponding to C is the T m-invariant decomposition of ?1 (C ) into the \moment-angle" blocks B (6.2) corresponding to the faces C of C . Thus, ma(C ) is de ned from the commutative diagram ma(C ) ? ?? (D2 )m ?!
? ? y ? ? y

:

C ? ?? I m ?! m acts on ma(C ) with orbit space C . The torus T Let K n?1 be a simplicial complex on the set m]. In section 4.2 two canonical cubical subcomplexes of I m , namely cub(K ) (4.3) and cc(K ) (4.4), were associated to K n?1 . We denote the corresponding moment-angle complexes by WK and ZK
respectively. Thus, we have

W

(6.3)

? ? y

K

? ?? (D2 )m ?!
? ? y

Z
and

? ? y

K

? ?? (D2 ) ?!
? ? y

m

;

cub(K ) ? ?? I m ?! cc(K ) ? ?? I m ?! where the horizontal arrows are embeddings, while the vertical ones are orbit maps for T m-actions. Note that dim ZK = m + n and dim WK = m + n ? 1. Remark. Suppose that K = KP for some simple polytope P . Then it follows from (6.1) that ZK is identi ed with ZP (or, more precisely, with ie (ZP )). The simple polyhedral complex PK , used in 48] to de ne ZK for general K , now can be interpreted as a certain face decomposition of the cubical complex cc(K ) (see also the proof of Lemma 6.13 below). Note the complex ZK depends on the ambient set m] of K as well as the complex K . In the case when it is important to emphasize this we will use the notation ZK; m]. If we assume that K is a simplicial complex on the vertex set m], then ZK is determined by K . However, in some situations (see e.g. section 6.4) it is convenient to consider simplicial complexes K on m] whose vertex sets are proper subsets of m]. Let fig be a ghost vertex of K , i.e. fig is a one-element subset of m] which is not a vertex of K . Then the whole cubical subcomplex cc(K ) I m is


6.3. CELL DECOMPOSITIONS OF MOMENT-ANGLE COMPLEXES

89

contained in the facet fyi = 1g of I m (see the remark after Construction 4.9). The following proposition follows easily from (6.3). Proposition 6.11. Suppose fi1 g; : : : ; fik g are ghost vertices of K . Then ZK; m] = ZK; m]nfi1 ;::: ;ik g T k : We call this easy observation \stabilization of moment-angle complexes via the multiplication by tori". It means that if we embed K into a set larger than its vertex set then the corresponding complex ZK is multiplied by the torus of dimension equal to the number of \ghost vertices". Example 6.12. 1. Let K be the boundary of (m ? 1)-simplex. Then cc(K ) is the union of m facets of I m meeting at the vertex (1; : : : ; 1), and ZK is the (2m ? 1)-sphere S 2m?1 (compare with Example 6.7). 2. Let K be an (m ? 1)-simplex. Then cc(K ) is the whole cube I m and ZK is the m-disc (D2 )m . Lemma 6.13. Suppose K is a simplicial (n?1)-sphere. Then ZK is an (m+n)dimensional (closed) manifold. Proof. In this proof we identify the polyhedrons jK j and j cone(K )j with their images cub(K ) I m and cc(K ) I m under the map j cone(K )j ! I m , see e Proposition 4.10. For each vertex fig 2 K denote by Fi the union of (n ? 1)-cubes e e e of cub(K ) that contain fig. Alternatively, Fi is j starK 0 figj. These F1 ; : : : ; Fm will e play the role of facets of a simple polytope. If K = KP for some P , then Fi is the image of a facet of P under the map iP : C (P ) ! I m (see Construction 4.5). As in the case of simple polytopes, we de ne \faces" of cc(K ) as non-empty intersections e e of \facets" F1 ; : : : ; Fm . Then the \vertices" (i.e. non-empty intersections of n \facets") are the barycenters of (n ? 1)-simplices of jK j. For every such barycenter b, denote by Ub the open subset of cc(K ) obtained by deleting all \faces" not containing b. Then Ub is identi ed with Rn , while ?1 (Ub ) is homeomorphic to + T m?n R2n . This de nes a structure of manifold with corners on the n-ball cc(K ) = j cone(K )j, with atlas fUb g. Furthermore, ZK = ?1 (cc(K )) is a manifold, with atlas f ?1 (Ub )g. Problem 6.14. Characterise simplicial complexes K for which ZK is a manifold. We will see below (Theorem 7.6) that if ZK is a manifold, then K is a Gorenstein* complex (see De nition 3.37) for homological reasons. Hence, the answer to the above problem is somewhere between \simplicial spheres" and \Gorenstein* complexes". Here we consider two cell decompositions of (D2 )m and apply them to construct cell decompositions for moment-angle complexes. The rst one has 5m cells and descends to a cell complex structure (with 5 types of cells) on any moment-angle complex ma(C ) (D2 )m . The second cell decomposition of (D2 )m has only 3m cells, but it de nes a cell complex structure (with 3 types of cells) only on momentangle complexes ZK . Let us consider the cell decomposition of D2 with one 2-cell D, two 1-cells I , T and two 0-cells 0, 1, shown on Figure 6.1 (a). It de nes a cell complex structure

6.3. Cell decompositions of moment-angle complexes


90

T

'$'$ ss s &%&%
6. MOMENT-ANGLE COMPLEXES

D

0

I

1

T

D

1

(a)

(b)

Figure 6.1. Cell decompositions of D2 .

on the poly-disc (D2 )m with 5m cells. Each cell of this complex is a product of cells of 5 di erent types: Di , Ii , 0i , Ti and 1i , i = 1; : : : ; m. We encode cells in the language of \sign vectors" used in the theory of hyperplane arrangements, see e.g. 22]. Each cell of (D2 )m with respect to our 5m-cell decomposition will be represented by a sign vector R 2 fD; I ; 0; T ; 1gm. We denote by RD , RI , R0 , RT and R1 respectively the D-, I -, 0-, T - and 1-component of R. Each of these components can be seen as a subset of m], and all ve subsets are complementary. This justi es the notations jRD j, jRI j, jR0 j, jRT j and jR1 j for the number of D-, I -, 0-, T - and 1-entries of R respectively. In particular, we see that the closure of a cell R is homeomorphic to a product of jRD j discs, jRI j segments and jRT j circles. Our rst observation is that this cell decomposition of (D2 )m induces a cell decomposition of any moment-angle complex in (D2 )m : Lemma 6.15. For any cubical subcomplex C of I m the corresponding momentangle complex ma(C ) is a cel lular subcomplex of (D2 )m . Proof. Indeed, ma(C ) is a union of \moment-angle" blocks B (6.2), and each B is the closure of cell R with RD = n , R0 = , RT = m] n , RI = R1 = ?. Now we restrict our attention to the moment-angle complex ZK corresponding to cubical complex cc(K ) I m (see (6.3)). By the de nition, ZK is the union of moment-angle blocks B (D2 )m with 2 K . Denote (6.4) B := B? = (z1 ; : : : ; zm ) 2 (D2 )m : jzj j = 1 for j 2 : = Then B = ?1 (C ) (remember our previous notation C := C? ) and B B for any . It follows that (6.5) ZK = B (compare this with the note after (4.4)). Remark. If K = KP for a simple polytope P and j j = n, then B is ie (Bv ) T for v = j2 Fj . Hence, (6.5) reduces to (6.1) in this case. Note that B \ B 0 = B \ 0 . This observation allows us to simplify the cell decomposition from Lemma 6.15 in the case ma(C ) = ZK . For this we replace the union of cells 0, I , D (see Figure 6.1 (a)) by one 2-dimensional cell (which we keep denoting D for simplicity). The resulting cell decomposition of D2 with 3 cells is shown on Figure 6.1 (b). It de nes a cell decomposition of (D2 )m with 3m cells,
2K


6.3. CELL DECOMPOSITIONS OF MOMENT-ANGLE COMPLEXES

91

each of which is a product of 3 di erent types Di , Ti and 1i , i = 1; : : : ; m. Again we use the sign vector language and encode the new cells of (D2 )m by sign vectors T 2 fD; T ; 1gm. The notation TD , jTT j etc. have the same meaning as in the case of 5m -cell decomposition. The closure of T is now a product of jTD j discs and jTT j circles. Lemma 6.16. The moment-angle complex ZK is a cel lular subcomplex of (D2 )m with respect to the 3m-cel l decomposition (see Figure 6.1 (b)). Those cel ls T (D2 )m which form ZK are determined by the condition TD 2 K . Proof. Since B = B? is the closure of cell T with TD = , TT = m] n and T1 = ?, the statement follows from (6.5). Remark. Note that for general C the moment-angle complex ma(C ) is not a cell subcomplex with respect to the 3m-cell decomposition of (D2 )m . Lemma 6.17. Let : K1 ,! K2 be an inclusion of simplicial complexes on the sets m1 ] and m2 ] respectively. Then it induces an equivariant cel lular map ma : ZK1 ! ZK2 of the corresponding moment-angle complexes. Proof. Assign the i-th vector of the standard basis of C m1 to the element i 2 m1 ], and similarly for C m2 and m2 ]. This allows us to extend the map : m1 ] ! m2 ] to an inclusion C : C m1 ! C m2 . For any subset m1 ] the map C takes B C m1 (see (6.4)) to B ( ) C m2 . Since is a simplicial map, for a simplex of K1 we have ( ) 2 K2 and C (B ) ZK2 (see (6.5)). Hence, C de nes an (equivariant) map ma : ZK1 ! ZK2 . Let us apply the construction of ZK (see (6.3)) to the case when K = ? (regarded as a simplicial complex on m]). Then cc(K ) = (1; : : : ; 1) 2 I m (the cone over the empty set is just one vertex), and so Z? = ?1 (1; : : : ; 1) = T m , where Z? = Z?; m]. (Another way to see this is to apply Proposition 6.11 in the case K = ?.) We also observe that Z? is contained in ZK for any K on m] as a T m-stable subset. Lemma 6.18. Let K be a simplicial complex on the vertex set m]. The inclusion Z? ,! ZK is a cel lular embedding homotopical to a map to a point, i.e. the torus Z? is a cel lular subcomplex contractible within ZK . Proof. Z? ZK is a cellular subcomplex since it is the closure of the mdimensional cell T with TT = m]. So it remains to prove that T m is contractible within ZK . We do this by induction on m. If m = 1 then the only option for K is K = 0 (0-simplex), so ZK = D2 (see Example 6.12.2) and Z?; 1] = S 1 is contractible in ZK . Now suppose that the vertex set of K is m]. The embedding under the question factors as (6.6) Z?; m] ,! ZK m?1] ; m] ,! ZK; m] ; where K m?1] is the maximal subcomplex of K on the vertex set m ? 1], see (2.1). By Proposition 6.11, Z?; m] = Z?; m?1] S 1 and ZK m?1] ; m] = ZK m?1] ; m?1] S 1 . By the inductive hypothesis we may assume that the embedding Z?; m?1] ZK m?1]; m?1] is null-homotopic, so the composite embedding (6.6) is homotopic to the map Z?; m?1] S 1 ! ZK; m] that sends Z?; m?1] to a point and S 1 to the closure of the cell (1; : : : ; 1; T ) ZK; m] (the latter is understood as a vector of letters D; T ; 1). But since fmg is a vertex of K , the complex ZK also contains the


92

6. MOMENT-ANGLE COMPLEXES

cell (1; : : : ; 1; D), so a disc D2 is patched to the closure of (1; : : : ; 1; T ). It follows that the whole map (6.6) is null-homotopic.
Corollary 6.19. For any simplicial complex K on the vertex set m] the moment-angle complex ZK is simply connected. Proof. Indeed, the 1-skeleton of our cellular decomposition of ZK is contained in the torus Z? , which is null-homotopic by Lemma 6.18.

constructions from section 2.2. In particular, we describe moment-angle complexes corresponding to joins and connected sums of simplicial complexes and interpret bistellar moves (see De nition 2.39) as certain surgery-like operations on momentangle complexes. Agreement. Let = fj1 ; : : : ; jk g be a subset of m]. In this section we will denote the moment-angle block B = (D2 )k T m?k (6.4) by D2k T m?k . The boundary of B is @ B = D2k?j2g T m?k+1 = S 2k?1 T m?k nf
j2

6.4. Moment-angle complexes corresponding to joins, connected sums and bistellar moves Here we study the behavior of moment-angle complexes ZK with respect to

(compare with Example 6.7). We denote @ B = S 2k?1 T m?k . Furthermore, for any partition m] = into three complementary subsets with j j = i, j j = j , j j = r, we will use the notation D2i S 2j?1 T r for the corresponding subset of (D2 )m . Construction 6.20 (moment-angle complex corresponding to join). Let K1 , K2 be simplicial complexes on the sets m1 ], m2 ] respectively, and K1 K2 the join of K1 and K2 (see Construction 2.9). Identify the cube I m1 +m2 with I m1 I m2 . Then, using (4.4), we calculate cc(K
1

K2) =

1

2K1 ; 2 2K

2

C

1

2

=
2K1

1

= Hence,

1

C

2K1 ; 2 2K
1

2 2

C
2K

1

C C
2

2

2

= cc(K1 ) cc(K2 ):

This can be thought as a generalization of Proposition 6.4 to arbitrary simplicial complexes. Construction 6.21 (moment-angle complexes and connected sums). Suppose we are given two pure (n ? 1)-dimensional simplicial complexes K1 , K2 on the sets m1 ], m2 ] respectively, and let K1 # K2 be their connected sum at some 1 and 2 . (Here K1 # K2 is considered as a simplicial complex on m1 + m2 ? n], with suitable identi cation 1 = 2 = , see Construction 2.12.) If we regard K1 as a simplicial complex on m1 + m2 ? n], then the corresponding moment-angle

Z

K1 K2 = ZK1

ZK2 :


6.4. JOINS, CONNECTED SUMS AND BISTELLAR MOVES

93

complex is ZK1 T m2 ?n (Proposition 6.11), where ZK1 = ZK1 ; m1 ] , and similarly b b for K2. Denote K1 := K1 n f 1 g and K2 := K2 n f 2 g. Then (6.7) ZK1 = ZK1 n (T m1 ?n D2n ); ZK2 = ZK2 n (D2n T m2 ?n ) b b 1 2 by (6.5). Now we see that (6.8) ZK1 #K2 = ZK1 T m2?n T m1?n ZK2 ; b b where the two pieces are glued along T m1?n S 2n?1 T m2?n = T m1?n S 2n?1 1 2 T m2?n , using the identi cation of 1 with 2 . Equivalently, ZK1 #K2 = D2j j T m1+m2 ?n?j j:
Example 6.22. Let K1 = K be a on m] and K2 = @ n (the boundary 2 K and consider the connected sum in @ n is irrelevant). Note that Z@ n

2K1 or 2K 6 =

2

and (6.7) that (6.9) ZK # @ n ? = ZK S 1 n T m?n D2n S 1 T m?n S2n?1 S1 (T m?n S 2n?1 D2 ): Thus, ZK # @ n is obtained by removing the \equivariant" handle T m?n D2n S 1 from ZK S 1 and then attaching T m?n S 2n?1 D2 along the boundary T m?n S 2n?1 S 1. As we mentioned above, the connected sum with the boundary of simplex is a bistellar 0-move. Other bistellar moves also can be interpreted as \equivariant surgery operations" on ZK . Construction 6.23 (equivariant surgery operations). Let K be an (n ? 1)dimensional pure simplicial complex on m], and let 2 K be an (n ? 1 ? k)-simplex (1 6 k 6 n ? 2) such that link is the boundary @ of a k-simplex that is not a face of K . Let K 0 be the complex obtained from K by applying the corresponding bistellar k-move, see De nition 2.39: ? (6.10) K 0 = K n ( @ ) (@ ) (note that due to our assumptions K 0 has the same number of vertices as K ). The moment-angle complexes corresponding to @ and @ are D2(n?k) 2(n?k)?1 D2(k+1) respectively (this follows from Example 6.12 and S 2k+1 and S Construction 6.20). Using stabilization arguments (Proposition 6.11), we obtain (6.11) ? ZK 0 = ZK n T m?n?1 D2(n?k) S 2k+1 (T m?n?1 S 2(n?k)?1 D2(k+1) ); where T m?n?1 S 2(n?k)?1 D2(k+1) is attached along its boundary T m?n?1 S 2(n?k)?1 S 2k+1 . This describes the behavior of ZK under bistellar k-moves (the cases k = 0 and k = n ? 1 are covered by (6.9)).

2n?1 S1 S (see examples 6.7 and 6.12), therefore Z@dn = S 2n?1

D2n S 1 S

pure (n ? 1)-dimensional simplicial complex of n-simplex). Choose a maximal simplex K # @ n (the choice of a maximal simplex = S 2n+1 can be decomposed as

2n?1

D2 D2 . Now it follows from (6.8)


94

6. MOMENT-ANGLE COMPLEXES

sphere obtained from K by applying a bistel lar k-move (6.10), 0 < k < n ? 1. 0 Then the corresponding moment-angle manifolds ZK and ZK are T m-equivariantly 0 is obtained from K by applying a 0-move, then ZK 0 is cobordant cobordant. If K to ZK S 1 . Proof. We give a proof for k-moves, k > 1. The case k = 0 is considered similarly. Consider the product U = ZK 0; 1] of ZK with a segment. De ne X = T m?n?1 D2(n?k) S 2k+1 and Y = T m?n?1 D2(n?k) D2(k+1) (the latter is a \solid equivariant handle"). Since X ZK and X @ Y , we can attach Y to U at X 1 ZK 1. Denote the resulting manifold (with boundary) by V , i.e. V = U X Y . Then it follows from (6.11) that @ V = ZK ZK 0 (here ZK comes from ZK 0 U , while ZK 1 is replaced by ZK 0 ). This concludes the proof.

Lemma 6.24. Let K = K n?1 be a simplicial sphere and K 0 the simplicial

Now we have the following topological corollary of Pachner's Theorem 2.40. Theorem 6.25. Let K n?1 be a P L sphere. Then for some p the moment-angle manifold ZK T p is equivariantly cobordant to S 2n+1 T m+p?n?1. This cobordism is realized by a sequence of equivariant surgeries. Proof. By Theorem 2.40, the P L sphere K is taken to @ n by a sequence of bistellar moves. Since Z@ n = S 2n+1 , the statement follows from Lemma 6.24.

arguments for the statements about the cohomology of quasitoric manifolds, which we left unproved in section 5.2. Let E T m be the contractible space of the universal principal T m-bundle over classifying space B T m. It is well known that B T m is (homotopy equivalent to) the product of m copies of in nite-dimensional pro jective space C P 1 . The cell decomposition of C P 1 with one cell in every even dimension determines the canonical cell decomposition of B T m. The cohomology of B T m (with coe cients in k) is thus the polynomial ring k v1 ; : : : ; vm ], deg vi = 2. Definition 6.26. Let X be a T m-space. The Borel construction (alternatively, homotopy quotient or associated bund le ) is the identi cation space E T m T m X := E T m X= ; where (e; x) (eg; g?1x) for any e 2 E T m, x 2 X , g 2 T m. The pro jection (e; x) ! e displays E T m T m X as the total space of a bundle m T m X ! B T m with bre X and structure group T m . At the same time, ET there is a principal T m-bundle E T m X ! E T m T m X . In the sequel we denote the Borel construction E T m T m X corresponding to m -space X by BT X . In particular, for any simplicial complex K on m vertices aT we have the Borel construction BT ZK and the bundle p : BT ZK ! B T m with bre ZK . For each i = 1; : : : ; m denote by B Ti the i-th factor in B T m = (C P 1 )m . For a subset m] we denote by B T the product of B Ti 's with i 2 . Obviously, B T is a cellular subcomplex of B T m, and B T = B T k if j j = k.

6.5. Borel constructions and Davis{Januszkiewicz space Here we study basic homotopy properties of ZK . We also provide necessary


6.5. BOREL CONSTRUCTIONS AND DAVIS{JANUSZKIEWICZ SPACE

95

subcomplex

Definition 6.27. Let K be a simplicial complex. We refer to the cellular

as the Davis{Januszkiewicz space , and denote it DJ (K ). The following statement is an immediate corollary of the de nition of Stanley{ Reisner ring k(K ) (De nition 3.1). Proposition 6.28. The cel lular cochain algebra C (DJ (K )) and the cohomology algebra H (DJ (K )) are isomorphic to the face ring k(K ). The cel lular inclusion i : DJ (K ) ,! B T m induces the quotient epimorphism i : k v1 ; : : : ; vm ] ! k(K ) = k v1 ; : : : ; vm]=IK in the cohomology. Theorem 6.29. The bration p : BT ZK ! B T m is homotopy equivalent to the cel lular inclusion i : DJ (K ) ,! B T m. More precisely, there is a deformation retraction BT ZK ! DJ (K ) such that the diagram

2K

BT

BT m

BT ZK ? ?? B T m ?p ! ?
? y

is commutative. Proof. Consider the decomposition (6.5). Since each B ZK is T m-stable, the Borel construction BT ZK = E T m T m ZK is patched from the Borel constructions E T m T m B for 2 K . Suppose j j = j ; then B = (D2 )j T m?j (see (6.4)). By the de nition of Borel construction, E T m T m B = (E T j T j (D2 )j ) E T m?j . The space E T j T j (D2 )j is the total space of a (D2 )j -bundle over B T j . It follows that there is a deformation retraction E T m T m B ! B T , which de nes a homotopy equivalence between the restriction of p : BT ZK ! B T m to E T m T m B and the cellular inclusion B T ,! B T m. These homotopy equivalences corresponding to di erent simplices 2 K t together to yield a required homotopy equivalence between p : BT ZK ! B T m and i : DJ (K ) ,! B T m.
Corollary 6.30. The moment-angle complex ZK is the homotopy bre of the cel lular inclusion i : DJ (K ) ,! B T m. As a corollary, we get the following statement, rstly proved in 48, Theorem 4.8]. Corollary 6.31. The cohomology algebra H (BT ZK ) is isomorphic to the face ring k(K ). The projection p : BT ZK ! B T m induces the quotient epimorphism p : k v1 ; : : : ; vm ] ! k(K ) = k v1 ; : : : ; vm ]=IK in the cohomology. Corollary 6.32. The T m-equivariant cohomology of ZK is isomorphic to the Stanley{Reisner ring of K : HT m (ZK ) = k(K ): The following information about the homotopy groups of ZK can be retrieved from the above constructions.

DJ (K ) ? ?? B T m ?i !


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6. MOMENT-ANGLE COMPLEXES

Theorem 6.33. (a) The complex ZK is 2-connected (i.e. 1 (ZK ) = 2 (ZK ) = 0), and i (ZK ) = i (BT ZK ) = i (DJ (K )) for i > 3. (b) If K = KP and P is q-neighborly (see De nition 1.15), then i (ZK ) = 0 for i < 2q + 1. Moreover, 2q+1 (ZP ) is a free Abelian group generated by the (q + 1)-element missing faces of KP . Proof. Note that B T m = K (Zm; 2) and the 3-skeleton of DJ (K ) coincides with that of B T m . If P is q-neighborly, then it follows from De nition 6.27 that the (2q + 1)-skeleton of DJ (KP ) coincides with that of B T m . Now, both statements follow easily from the exact homotopy sequence of the map i : DJ (K ) ! B T m with homotopy bre ZK (see Corollary 6.30). Remark. We say that a simplicial complex K on the set m] is k-neighborly if any k-element subset of m] is a simplex of K . (This de nition is an obvious extension of the notion of k-neighborly simplicial polytope to arbitrary simplicial complexes.) Then the second part of Theorem 6.33 holds for arbitrary q-neighborly simplicial complex. Suppose now that K = KP for some simple n-polytope P and M 2n is a quasitoric manifold over P with characteristic function ` (see De nition 5.10). Then we have the subgroup H (`) T m acting freely on ZP and the principal T m?n-bundle ZP ! M 2n (Proposition 6.5). Proposition 6.34. The Borel construction E T n T n M 2n is homotopy equivalent to BT ZP . Proof. Since H (`) acts freely on ZP , we have BT ZP = E T m T m ZP ? = E H (`) E T m =H (`) T m =H (`) ZP =H (`) ' E T n T n M 2n : Corollary 6.35. The T n-equivariant cohomology ring of a quasitoric manifold M 2n over P n is isomorphic to the Stanley{Reisner ring of P n : HT n (M 2n ) = k(P n ): Proof. It follows from Proposition 6.34 and Corollary 6.32. Theorem 6.36 ( 48, Theorem 4.12]). The Leray{Serre spectral sequence of the

bund le (6.12) E T n T n M 2n ! B T n p;q p;q with bre M 2n col lapses at the E2 term, i.e. E2 = E1 . Proof. Since both B T n and M 2n have only even-dimensional cells (see Proposition 5.16), all the di erentials in the spectral sequence are trivial by dimensional reasons. Corollary 6.37. Projection (6.12) induces a monomorphism k t1 ; : : : ; tn ] ! k(P ) in the cohomology. The inclusion of bre M 2n ,! E T n T n M 2n induces an epimorphism k(P ) ! H (M 2n ). Now we are ready to give proofs for the statements from section 5.2.


6.6. WALK AROUND THE CONSTRUCTION OF ZK

97

Proof of Lemma 5.17 and Theorem 5.18. The monomorphism H (B T n) = k t1 ; : : : ; tn ] ! k(P ) = H (E T n T n M 2n ) takes ti to i , i = 1; : : : ; n. By Theorem 6.36, k(P ) is a free k t1 ; : : : ; tn ]-module, hence, 1 ; : : : ; n is a regular sequence. Therefore, the kernel of k(P ) ! H (M 2n ) is exactly J` = ( 1 ; : : : ; n ).

Many of our previous constructions (namely, the cubical complex cc(K ), the moment-angle complex ZK , the Borel construction BT ZK , the Davis{Januszkiewicz space DJ (K ), and also the complement U (K ) of a coordinate subspace arrangement appearing in section 8.2) admit a unifying combinatorial interpretation in terms of the following construction, which was mentioned to us by N. Strickland (in private communications). Construction 6.38. Let X be a space, and W a subspace of X . Let K be a simplicial complex on the set m]. De ne the following subset in the product of m copies of X : Y Y X W: K (X ; W ) =
Example 6.39. 1. cc(K ) = K (I 2. ZK = K (D2 ; S 1) (see (6.5)). 3. DJ (K ) = K (C P 1 ; ) (see De 4. BT ZK = K (E S 1 S1 D2 ; E S 1
1

6.6. Walk around the construction of ZK : generalizations, analogues and additional comments

2K i2

; 1) (see (4.4)).

i2 =

nition 6.27). S 1 S 1 ) (see the proof of Theorem 6.29). Another unifying description of the above spaces can be achieved using categorical constructions of limits and colimits of di erent diagrams over the face category cat(K ) of K . (The ob jects of cat(K ) are simplices 2 K and the morphisms are inclusions.) For instance, De nition 6.27 is an example of this procedure: the Davis{Januszkiewicz space is the colimit of the diagram of spaces over cat(K ) that assigns B T to a simplex 2 K . In the case when K is a ag complex (see Definition 2.18 and Proposition 2.19) the colimit over cat(K ) reduces to the graph product , studied in the theory of groups (see e.g. 43]). Well-known examples of graph products include right-angled Coxeter and Artin groups (see e.g. 48], 49]). The most general categorical setup for the above constructions involves the notion of homotopy colimit 24], 138]. This fundamental algebraic-topological concept has already found combinatorial applications, see 139]. The complex ZK can be seen as the homotopy colimit of a certain diagram of tori; this interpretation is similar to the homotopy colimit description of toric varieties proposed in 139]. For more information on this approach see 113]. The combinatorial theory of toric spaces is parallel to some extent to its Z=2-, or \real", counterpart. We say a few words about the Z=2-theory here, referring the reader to 48], 49] and other papers of R. Charney, M. Davis, T. Januszkiewicz and their co-authors for a more detailed treatment (some further results can be also found in 113]). The rst step is to pass from the torus T m to its \real analogue", the group (Z=2)m. The standard cube I m = 0; 1]m is the orbit space for the action of (Z=2)m on the bigger cube ?1; 1]m, which in turn can be regarded as a \real analogue" of the poly-disc (D2 )m C m . Now, given a cubical subcomplex


98

6. MOMENT-ANGLE COMPLEXES

proof is similar to that of Lemma 6.13). Thereby, for any simplicial sphere K n?1 with m vertices we get a (Z=2)m-symmetric n-manifold with a (Z=2)m-invariant cubical subdivision. As suggested by the results of 10], this class of cubical manifolds may be useful in the combinatorial theory of face vectors of cubical complexes (see section 4.1). The real analogue RZP of the manifold ZP (corresponding to the case of a polytopal simplicial sphere) is the universal Abelian cover of the polytope P n regarded as an orbifold (or manifold with corners), see e.g. 68, x4.5]. In 78] manifolds RZP and ZP are interpreted as the con guration spaces of equivariant hinge mechanisms (or linkages ) in R2 and R3 . 2 2 Example 6.40. Let Pm be an m-gon. Then RZPm is a 2-dimensional manifold. It is easy to see that RZP32 = RZ 2 = S 2 (a 2-sphere patched from 8 triangles) and RZP42 = RZ 1 1 = T 2 (a 2-torus patched from 16 squares). More generally, 2 RZPm is patched from 2m polygons, meeting by 4 at each vertex. Hence, we have m2m?2 vertices and m2m?1 edges, so the Euler characteristic is 2 (RZPm ) = 2m?2(4 ? m): 2 Thus, RZPm is a surface of genus 1 ? 2m?1 + m2m?3. This also can be seen directly 2 by decomposing Pm into a connected sum of an (m ? 1)-gon and triangle and using the real version of Example 6.22. Replacing T n by (Z=2)n in De nition 5.10, we obtain a real version of quasitoric manifolds, which was introduced in 48] under the name smal l covers . Thereby a small cover of a simple polytope P n is a (Z=2)n-manifold M n with quotient P n . The name refers to the fact that any branched cover of P n (as an orbifold) by a smooth manifold has at least 2n sheets. Small covers were studied in 48] along with quasitoric manifolds, and many results on quasitoric manifolds quoted from 48] in section 5.2 have analogues in the small cover case. Also, like in the torus case, every small cover is the quotient of the universal cover RZP by a free action of the group (Z=2)m?n. An important class of small covers (and quasitoric manifolds) was introduced in 48, Example 1.15] under the name pul lbacks from the linear model . They correspond to simple polytopes P n whose dual triangulation can be folded onto the (n ? 1)-simplex (more precisely, the polytopal sphere KP admits a non-degenerate simplicial map onto n?1 ; note that this is always the case when KP is a barycentric subdivision of some other polytopal sphere, see Example 2.15). If this condition is satis ed then there exists a special characteristic map (5.4) which assigns to each facet of P n a coordinate subtorus Ti T n (or coordinate subgroup (Z=2)i (Z=2)n in the small cover case). Pullbacks from the linear model have a number of nice properties, in particular, they are all stably parallelizable ( 48, Corollary 6.10], compare with Theorem 5.33). The existence of a non-degenerate simplicial map

RZK = K ?1; 1]; f?1; 1g : This cubical complex was studied, e.g. in 10] under the name mirroring construction . If K is a simplicial (n?1)-sphere, then RZK is an n-dimensional manifold (the

Construction 6.38 we have

C I m , one can construct a (Z=2)m-symmetrical cubical complex embedded into ?1; 1]m just in the same way as it is done in De nition 6.10. In particular, for any simplicial complex K on the vertex set m] one can introduce the real versions RZK and RWK of the moment-angle complexes ZK and WK (6.3). In the notations of
?


6.6. WALK AROUND THE CONSTRUCTION OF ZK

99

from KP to n?1 can be reformulated by saying that the polytope P n admits a regular n-paint coloring . The latter means that the facets of P n can be colored with n paints in such a way that any two adjacent facets have di erent color. A simple polytope P n admits a regular n-paint coloring if and only if every 2-face has an even number of edges. This is a classical result for n = 3; the proof in the general case can be found in 81]. Some additional results about pullbacks from the linear model in dimension 3 were obtained in 79]. It was shown there that any small cover M 3 which is a pullback from the linear model admits an equivariant embedding into R4 = R3 R with the standard action of (Z=2)3 on R3 and the trivial action on R. Another result from 79] says that any such M 3 can be obtained from a set of 3-dimensional tori by applying several equivariant connected sums and equivariant Dehn twists (compare with section 6.4). Although not every simple 3-polytope admits a regular 3-paint coloring, a regular 4-paint coloring can always be achieved due to the Four Color Theorem. This argument was used in 48, Example 1.21] to prove that there is a small cover (or quasitoric manifold) over every simple 3-polytope. (One can just construct a characteristic map by assigning the coordinate circles in T 3 to the rst three colors and the diagonal circle to the fourth one.) On the other hand, it would be particularly interesting to develop a quaternionic analogue of the theory. Unlike the real case, not much is done here. To begin, of course, we have to replace T n by the quaternionic torus S p(1)n = (S 3 )n . Developing quaternionic analogues of toric and quasitoric manifolds is quite tricky. R. Scott in 119] used the quaternionic analogue of characteristic map to approach this problem. However, the non-commutativity of the quaternionic torus implies that it does not contain su ciently many subgroups for the resulting quaternionic toric manifolds to have an actual S p(1)n -action. A polytopal structure also appears in the quotients of some other types of manifolds studied in the quaternionic geometry, see e.g. 25]. We also mention that since only coordinate subgroups of T m are involved in the de nition of the moment-angle complex ZK , this particular construction of a toric space does have a quaternionic analogue which is an S p(1)m -space. At the end we give one example which builds on a generalization of the construction of ZK to the case of an arbitrary group G. Example 6.41 (classifying space for group G). Let K be a simplicial complex on the vertex set m]. Set ZK (G) := K (cone(G); G) (see Construction 6.38), where cone(G) is the cone over G with the obvious G-action. By the construction, the group Gm acts on ZK (G), with quotient cone(K ). It is also easy to observe that the diagonal subgroup in Gm acts freely on ZK (G), thus identifying ZK (G) as a principal G-space. Suppose now that K1 K2 Ki is a sequence of embedded simplicial complexes such that Ki is i-neighborly. The group G acts freely on the contractible space lim ZKi (G), and the corresponding quotient is thus the classifying ?! space B G. Thus, we have the following ltration in the universal bration E G ! B G: ZK1 (G) ,! ZK2 (G) ,! ,! ZKi (G) ,!

: The well-known Milnor ltration in the universal bration of the group G corresponds to the case Ki = i?1 .

# # ZK1 (G)=G ,! ZK2 (G)=G ,!

,! ZKi (G)=G ,!

#


100

6. MOMENT-ANGLE COMPLEXES


CHAPTER 7

Cohomology of moment-angle complexes and combinatorics of triangulated manifolds
quence of great importance for algebraic topology. This spectral sequence can be considered as an extension of Adams' approach to calculating the cohomology of loop spaces 1]. In 1960-70s di erent applications of the Eilenberg{Moore spectral sequence led to many important results on the cohomology of loop spaces and homogeneous spaces for Lie group actions. In this chapter we discuss some new applications of this spectral sequence to combinatorial problems. This section contains the necessary information about the spectral sequence; we follow L. Smith's paper 121] in this description. The following theorem provides an algebraic setup for the Eilenberg{Moore spectral sequence. Theorem 7.1 (Eilenberg{Moore 121, Theorem 1.2]). Let A be a commutative di erential graded k-algebra, and M , N di erential graded A-modules. Then there exists a spectral sequence fEr ; dr g converging to TorA (M ; N ) and whose E2 term is ? ? E2 i;j = Tor?i;j ] H M ]; H N ] ; i; j > 0; HA where H ] denotes the cohomology algebra (or module). The above spectral sequence lives in the second quadrant and its di erentials dr add (r; 1 ? r) to bidegree, r > 1. It is called the (algebraic) Eilenberg{Moore spectral sequence . For the corresponding decreasing ltration fF ?p TorA (M ; N )g in TorA (M ; N ) we have
? E1p;n+p = F ?p
X

7.1. The Eilenberg{Moore spectral sequence In their paper 60] of 1966, Eilenberg and Moore constructed a spectral se-

Topological applications of Theorem 7.1 arise in the case when A; M ; N are singular (or cellular) cochain algebras of certain topological spaces. The classical situation is described by the commutative diagram

?i+j=n

Tor?i;j (M ; N ) A

F ?p+1

X

?i+j=n

? TorAi;j (M ; N ) :

E ? ?? E ?!
? ? y

(7.1)

? ? y

0

B ? ?? B0 ; ?! where E0 ! B0 is a Serre bre bundle with bre F over a simply connected base B0 , and E ! B is the pullback along a continuous map B ! B0 . For any space X , let C (X ) denote either the singular cochain algebra of X or (in the case when X is a
101


102

7. COHOMOLOGY OF MOMENT-ANGLE COMPLEXES

cellular complex) the cellular cochain algebra of X . Obviously, C (E0 ) and C (B ) are C (B0 )-modules. Under these assumptions the following statement holds. Lemma 7.2 ( 121, Proposition 3.4]). TorC (B0 ) (C (E0 ); C (B )) is an algebra in a natural way, and there is a canonical isomorphism of algebras ? TorC (B0 ) C (E0 ); C (B ) ! H (E ): Applying Theorem 7.1 in the case A = C (B0 ), M = C (E0 ), N = C (B ) and taking into account Lemma 7.2, we come to the following statement. Theorem 7.3 (Eilenberg{Moore). There exists a spectral sequence of commutative algebras fEr ; dr g with (a) Er ) H (E ); ? ? (b) E2 i;j = Tor?i;jB0 ) H (E0 ); H (B ) . H( The spectral sequence of Theorem 7.3 is called the (topological) Eilenberg{ Moore spectral sequence . The case when B in (7.1) is a point is particularly important for applications, so we state the corresponding result separately. Corollary 7.4. Let E ! B be a bration over a simply connected space B with bre F . Then there exists a spectral sequence of commutative algebras fEr ; dr g with (a) Er ) H (F ); ? (b) E2 = TorH (B) H (E ); k . We refer to the spectral sequence of Corollary 7.4 as the Eilenberg{Moore spectral sequence of bration E ! B . Example 7.5. Let M 2n be a quasitoric manifold over P n (see De nition 5.10). Consider the Eilenberg{Moore spectral sequence of the bundle E T n T n M 2n ! B T n with bre M 2n . By Proposition 6.34, H (E T n T n M 2n ) = H (BT ZP ) = k(P n). The monomorphism k t1 ; : : : ; tn] = H (BT n) ! H (E T n T n M 2n) = k(P n ) takes ti to i (i = 1; : : : ; n), see (5.6). The E2 term of the Eilenberg{Moore spectral sequence is ? ? E2 ; = TorH; (BT n ) H (E T n T n M 2n ); k = Tork; t1 ;::: ;tn ] k(P n ); k : Since k(P n ) is a free k t1 ; : : : ; tn ]-module, we have ? ? Tork; t1;::: ;tn] k(P n ); k = Tor0; t1 ;::: ;tn ] k(P n ); k k = k(P n ) k t1;::: ;tn] k = k(P n )=( 1 ; : : : ; n ): 0 ? Therefore, E2 ; = k(P n )=J` and E2 p; = 0 for p > 0. It follows that the Eilenberg{ Moore spectral sequence collapses at the E2 term and H (M 2n ) = k(P n )=J` , in accordance with Theorem 5.18. Here we apply the Eilenberg{Moore spectral sequence to calculating the cohomology algebra of the moment-angle complex ZK . As an immediate corollary we obtain that the cohomology algebra inherits a canonical bi grading from the spectral sequence. The corresponding bigraded Betti numbers coincide with important combinatorial invariants of K introduced by Stanley 128].

7.2. Cohomology algebra of Z

K


7.2. COHOMOLOGY ALGEBRA OF ZK

103

Theorem 7.6. The fol lowing isomorphism of algebras holds:

This formula either can be seen as an isomorphism of graded algebras, where the grading in the right hand side is by the total degree, or used to de ne a bigraded algebra structure in the left hand side. In particular,

H (ZK ) = Tork v1 ;::: ;vm ] k(K ); k :

?

H p (ZK ) =
tative square (7.2)

X

?i+2j=p

? 2j Tork i;1 ;::: ;v v

m]

?

k(K ); k :

Proof. Let us consider the Eilenberg{Moore spectral sequence of the commu-

E

? ? y

? ?? E T ?!

? ? y

m

DJ (K ) ? ?? B T m; ?i ! where the left vertical arrow is the pullback along i. Corollary 6.30 shows that E is homotopy equivalent to ZK . By Proposition 6.28, the map i : DJ (K ) ,! B T m induces the quotient epimorphism i : C (B T m ) = k v1 ; : : : ; vm ] ! k(K ) = C (DJ (K )); where C ( ) denotes the cellular cochain algebra. Since E T m is contractible, there is a chain equivalence C (E T m) ' k. More precisely, C (E T m) can be identi ed with the Koszul resolution u1 ; : : : ; um ] k v1 ; : : : ; vm ] of k (see Example 3.24). Therefore, we have an isomorphism

(7.3)

TorC (BT m ) C DJ (K ) ; C (E T m ) = Tork v1 ;::: ;vm ] k(K ); k : The Eilenberg{Moore spectral sequence of commutative square (7.2) has
?

?

?

E2 = TorH (BT m ) H DJ (K ) ; H (E T m ) and converges to TorC (BT m ) (C (DJ (K )); C (E T m)) (Theorem 7.1). Since
TorH (BT m ) H DJ (K ) ; H (E T m) = Tork v1 ;::: ;vm ] k(K ); k ; it follows from (7.3) that the spectral sequence collapses at the E2 term, that is, E2 = E1 . Lemma 7.2 shows that the module TorC (BT m ) (C (DJ (K )); C (E T m )) is an algebra isomorphic to H (ZK ), which concludes the proof. Theorem 7.6 displays the cohomology of ZK as a bi graded algebra and says that the corresponding bigraded Betti numbers b?i;2j (ZK ) coincide with that of k(K ), see (3.5). The next theorem follows from Lemma 3.29 and Corollary 3.30. Theorem 7.7. The fol lowing isomorphism of bigraded algebras holds: H ; (ZK ) = H u1; : : : ; um] k(K ); d ; where the bigraded structure and the di erential in the right hand side are de ned by (3.4).
? ?


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7. COHOMOLOGY OF MOMENT-ANGLE COMPLEXES

In the sequel, given two subsets = fi1; : : : ; ip g, = fj1 ; : : : ; jq g of m], we will denote the square-free monomial ui1 : : : uip vj1 : : : vjq 2 u1 ; : : : ; um ] k(K ) by u v . Note that bideg u v = (?p; 2(p + q)). Remark. Since the di erential d in (3.4) does not change the second degree, the di erential bigraded algebra u1 ; : : : ; um ] k(K ); d] splits into the sum of di erential subalgebras consisting of elements of xed second degree. Corollary 7.8. The Leray{Serre spectral sequence of the principal T m-bund le m ZK ! BT ZK col lapses at the E3 term. ET Proof. The spectral sequence under consideration converges to H (E T m ZK ) = H (ZK ) and has E2 = H (T m) H (BT ZK ) = u1; : : : ; um] k(K ): The di erential in the E2 term acts as in (3.4). Hence, E3 = H E2 ; d] = H u1; : : : ; um] k(K ) = H (ZK ); by Theorem 7.7.

u1 ; : : : ; um ] k(K ) spanned by monomials u and u v such that is a simplex of K , j j = q and \ = ?. De ne A (K ) =
m M q=0

Construction 7.9. Consider the subspace A?q (K )

A?q (K ):

Since d(ui ) = vi and d(vi ) = 0, we have d(A?q (K )) A?q+1 (K ). Therefore, A (K ) is a cochain subcomplex in u1; : : : ; um] k(K ); d]. Moreover, A (K ) inherits the bigraded module structure from u1; : : : ; um ] k(K ), with di erential d adding (1; 0) to bidegree. Hence, we have an additive inclusion (i.e. a monomorphism of bigraded modules) ia : A (K ) ,! u1 ; : : : ; um ] k(K ). On the other hand, A (K ) is an algebra in the obvious way, but is not a subalgebra of u1 ; : : : ; um ] k(K ). 2 2 (For instance, v1 = 0 in A (K ), but v1 6= 0 in u1; : : : ; um] k(K ).) Nevertheless, we have multiplicative projection (an epimorphism of bigraded algebras) jm : u1 ; : : : ; um] k(K ) ! A (K ). The additive inclusion ia and the multiplicative pro jection jm satisfy jm ia = id. Lemma 7.10. The cochain complexes u1 ; : : : ; um] k(K ); d] and A (K ); d] are cochain homotopy equivalent and therefore have the same cohomology. This implies the fol lowing isomorphism of bigraded k-modules: ? H A (K ); d] = Tork v1 ;::: ;vm ] k(K ); k : Proof. A routine check shows that the cochain homotopy operator s for the Koszul resolution (see the proof of Proposition VII.2.1 in 92]) establishes a cochain homotopy equivalence between the maps id and ia jm from the algebra u1; : : : ; um] k(K ); d] to itself. That is, ds + sd = id ? ia jm :


7.2. COHOMOLOGY ALGEBRA OF ZK

105

We just illustrate the above identity on few simple examples. 1) s(u1 v2 ) = u1u2 ; ds(u1 v2 ) = u2 v1 ? u1v2 ; sd(u1 v2 ) = u1 v2 ? u2 v1 ; hence, (ds + sd)(u1 v2 ) = 0 = (id ? ia jm )(u1 v2 ); 2 2) s(u1 v1 ) = u2 = 0; ds(u1 v1 ) = 0; d(u1 v1 ) = v1 ; sd(u1 v1 ) = u1 v1 ; 1 hence, (ds + sd)(u1 v1 ) = u1 v1 = (id ? ia jm )(u1 v1 ); 2 2 2 2 3) s(v1 ) = u1 v1 ; ds(v1 ) = v1 ; d(v1 ) = 0; 2 2 2 hence, (ds + sd)(v1 ) = v1 = (id ? ia jm )(v1 ): Now we recall our cell decomposition of ZK , see Lemma 6.16. The cells of ZK are the sign vectors T 2 fD; T ; 1gm with TD 2 K . Assign to each pair ; of disjoint subsets of m] the vector T ( ; ) with T ( ; )D = , T ( ; )T = . Then T ( ; ) is a cell of ZK if and only if 2 K . Let C (ZK ) and C (ZK ) denote the cellular chain and cochain complexes of ZK respectively. Both complexes C (ZK ) and A (K ) have the same cohomology H (ZK ). The complex C (ZK ) has the canonical additive basis consisting of cochains T ( ; ) . As an algebra, C (ZK ) is

generated by the cochains Di , Tj (of dimension 2 and 1 respectively) dual to the cells Di = T (fig; ?) and Tj = T (?; fj g), 1 6 i; j 6 m. At the same time, A (K ) is multiplicatively generated by vi , uj , 1 6 i; j 6 m. Theorem 7.11. The correspondence v u 7! T ( ; ) establishes a canonical isomorphism between the di erential graded algebras A (K ) and C (ZK ). Proof. It follows directly from the de nitions of A (K ) and C (ZK ) that the proposed map is an isomorphism of graded algebras. So it remains to prove that it commutes with di erentials. Let d, dc and @c denote the di erential in A (K ), C (ZK ) and C (ZK ) respectively. Since d(vi ) = 0 and d(ui ) = vi , we need to show that dc (Di ) = 0, dc (Ti ) = Di . We have @c(Di ) = Ti , @c (Ti ) = 0. A 2-cell of ZK is either Dj or Tjk = Tj Tk (k 6= j ). Then hdc Ti ; Dj i = hTi ; @c Dj i = hTi ; Tj i = ij ; hdc Ti ; Tjk i = hTi ; @c Tjk i = 0; where ij = 1 if i = j and ij = 0 otherwise. Hence, dc (Ti ) = Di . Further, a 3-cell of ZK is either Dj Tk or Tj1 j2 j3 = Tj1 Tj2 Tj3 . Then hdc Di ; Dj Tk i = hDi ; @c (Dj Tk )i = hDi ; Tjk i = 0; hdc Di ; Tj1 j2 j3 i = hDi ; @c Tj1 j2 j3 i = 0: Hence, dc (Di ) = 0. The above theorem provides a topological interpretation for the di erential algebra A (K ); d]. In the sequel we will not distinguish the cochain complexes A (K ) and C (ZK ), and identify ui with Ti , vi with Di . Now we can summarise the results of Proposition 3.4, Lemma 3.32, Lemma 6.17, Corollary 6.32 and Theorem 7.11 in the following statement describing the functorial properties of the correspondence K 7! ZK . Proposition 7.12. Let us introduce the fol lowing functors: Z , the covariant functor K 7! ZK from the category of nite simplicial complexes and simplicial inclusions to the category of toric spaces and equivariant maps (the moment-angle complex functor);


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k( ), the contravariant functor K 7! k(K ) from simplicial complexes to graded k-algebras (the Stanley{Reisner functor); Tor-alg, the contravariant functor ? K 7! Tork v1 ;:::;vm ] k(K ); k from simplicial complexes to bigraded k-algebras (the Tor-algebra functor, it coincides with the composition of k( ) and Tork v1 ;:::;vm ] ( ; k)); HT , the contravariant functor X 7! HT (X ) from the category of toric spaces and equivariant maps to k-algebras (the equivariant cohomology functor); H , the contravariant functor X 7! H (X ) from spaces to k-algebras (the ordinary cohomology functor). Then we have the fol lowing identities: HT Z = k( ); H Z = Tor-alg: The later identity implies that for every simplicial inclusion : K1 ! K2 the cohomology map ma : H (ZK2 ) ! H (ZK1 ) coincides with the induced homomorphism t (3.7) of Tor-algebras. In particular, induces a homomorphism H ?q;2p (ZK2 ) ! H ?q;2p (ZK1 ) of bigraded cohomology modules. In the Cohen{Macaulay case we have the following reduction theorem for the cohomology of ZK . Theorem 7.13. Suppose that K n?1 is Cohen{Macaulay, and let J be an ideal in k(K ) generated by degree-two regular sequence of length n. Then the fol lowing isomorphism of algebras holds: ? H (ZK ) = Tork v1 ;::: ;vm ]=J k(K )=J ; k : Proof. This follows from Theorem 7.6 and Lemma 3.35.
Note that the k-algebra k(K )=J is nite-dimensional (unlike k(K )). In some circumstances (see section 7.4) this helps to calculate the cohomology of ZK more e ciently. The bigraded structure in the algebra A (K ); d] de nes a bigrading in the cellular chain complex C (ZK ); @c ] via the isomorphism of Theorem 7.11. We have (7.4) bideg(Di ) = (0; 2); bideg(Ti ) = (?1; 2); bideg(1i ) = (0; 0): The di erential @c adds (?1; 0) to bidegree and thus the bigrading descends to the cellular homology of ZK . In this section we assume that the ground eld k is of zero characteristic. De ne the bigraded Betti numbers (7.5) b?q;2p (ZK ) = dim H?q;2p C (ZK ); @c ; q; p = 0; : : : ; m: Theorem 7.11 and Lemma 7.10 show that ? ? p (7.6) b?q;2p (ZK ) = dim Tor?qv;12;::: ;vm ] k(K ); k = ?q;2p k(K ) ; k

7.3. Bigraded Betti numbers of ZK : the case of general K


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107

see (3.5). Alternatively, b?q;2p (ZK ) equals the dimension of (?q; 2p)-th bigraded component of the cohomology algebra H u1; : : : ; um] k(K ); d]. For the ordinary Betti numbers bk (ZK ) we have (7.7)

bk (ZK ) =

X

?q+2p=k

b?q;2p (ZK ); k = 0; : : : ; m + n:

The lemma below describes some basic properties of bigraded Betti numbers (7.5).
edges, (a) (b) (c) (d) (e) (f )
Lemma 7.14. Let K n?1 be a simplicial complex with m = f0 vertices and f1 and ZK the corresponding moment-angle complex, dim ZK = m + n. Then b0;0 (ZK ) = b0 (ZK ) = 1 and b0;2p (ZK ) = 0 for p > 0; b?q;2p = 0 for p > m or q > p; b1 (ZK ) = b2 (ZK ) = 0; ? b3 (ZK ) = b?1;4 (ZK ) = f20 ? f1 ; b?q;2p (ZK ) = 0 for q > p > 0 or p ? q > n; bm+n (ZK ) = b?(m?n);2m(ZK ).

Proof. In this proof we calculate the Betti numbers using the cochain subcomplex A (K ) u1; : : : ; um] k(K ). The module A (K ) has the basis consisting of monomials u v with 2 K and \ = ?. Since bideg vi = (0; 2), bideg uj = (?1; 2), the bigraded component A?q;2p (K ) is spanned by monomials u v with j j = p ? q and j j = q. In particular, A?q;2p (K ) = 0 if p > m or q > p, whence the assertion (b) follows. To prove (a) we observe that A0;0 (K ) is generated by 1, while any v 2 A0;2p (K ) (p > 0) is a coboundary, whence H 0;2p (ZK ) = 0 for p > 0. Now look at assertion (e). Every u v 2 A?q;2p (K ) has 2 K , while any simplex of K has at most n vertices. It follows that A?q;2p (K ) = 0 for p ? q > n. By (b), b?q;2p (ZK ) = 0 for q > p, so it remains to prove that b?q;2q (ZK ) = 0 for q > 0. The module A?q;2q (K ) is generated by monomials u with j j = q. Since d(ui ) = vi , it follows easily that there are no non-zero cocycles in A?q;2q (K ). Hence, H ?q;2q (ZK ) = 0. The assertion (c) follows from (e) and (7.7). It also follows from (e) that H 3 (ZK ) = H ?1;4 (ZK ). The basis for A?1;4 (K ) consists of monomials uj vi , i 6= j . We have d(uj vi ) = vi vj and d(ui uj ) = uj vi ? ui vj . Hence, uj vi is a cocycle if and only if fi; j g is not a 1-simplex in K ; in this case two cocycles uj vi and ui vj represent the same cohomology class. Assertion (d) follows. The remaining assertion (f ) follows from the fact that a monomial u v 2 A (K ) has maximal total degree (m + n) if and only if j j = n and j j = m ? n.

Lemma 7.14 shows that non-zero bigraded Betti numbers br;2p (ZK ), r 6= 0 appear only in the strip bounded by the lines p = m, r = ?1, p + r = 1 and p + r = n in the second quadrant, see Figure 7.1 (a). The homogeneous component C?q;2p (ZK ) has basis of cellular chains T ( ; ) with 2 K , j j = p ? q and j j = q. It follows that (7.8)
? dim C?q;2p (ZK ) = fp?q?1 m?qp+q ;


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2

m

2

m

. . .
4 2

. . .
4 2

?m

?1 0 ?(m ? n) (a) arbitrary K n?1

0

?m

?1 0 ?(m ? n) (b) jK j = S n?1

0

Figure 7.1. Possible locations of non-zero bigraded Betti numbers b?q;2p (ZK ) (marked by ).

where (f0 ; f1; : : : ; fn?1 ) is the f -vector of K n?1 and f?1 = 1. The di erential @c does not change the second degree: @c : C?q;2p (ZK ) ! C?q?1;2p (ZK ): Hence, the chain complex C ; (ZK ) splits as follows:

C ; (ZK ); @c ] =

m M p=0

C ;2p (ZK ); @c :

Remark. The similar decomposition holds also for the cellular cochain complex C ; (ZK ); dc ] = A ; (K ); d]. Let us consider the Euler characteristic of complex C ;2p (ZK ); @c ]:

(7.9)

p (ZK ) :=

m X q=0

(?1)q dim C?q;2p (ZK ) = (ZK ; t) =
m X p=0

m X q=0

(?1)q b?q;2p (ZK ):

De ne the generating polynomial (ZK ; t) by

p (ZK )t2p

:

The following theorem calculates this polynomial in Theorem 7.15. For every (n ? 1)-dimensional vertices it holds that (7.10) (ZK ; t) = (1 ? t2 )m?n (h0 + h1 t2 + + hn where (h0 ; h1 ; : : : ; hn ) is the h-vector of K . Proof. It follows from (7.9) and (7.8) that (7.11)
p (ZK ) = m X

terms of the h-vector of K . simplicial complex K with m
? t2n ) = (1 ? t2 )m F k(K ); t ;

? (?1)p?j fj?1 m? jj ; p j =0


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109

Then (7.12) (ZK ; t) =

m X p=0

p (K )t2p = m X j =0

mm XX

? t2j t2(p?j) (?1)p?j fj?1 m? jj p p=0 j =0
n X j =0

=

fj?1 t2j (1 ? t2 )m?j = (1 ? t2 )m :

fj?1 (t?2 ? 1)?j :

Denote h(t) = h0 + h1 t + + hn tn . From (1.7) we get n X tn h(t?1 ) = (t ? 1)n fi?1 (t ? 1)?i i=0

Substituting t?2 for t above, we nally rewrite (7.12) as (ZK ; t) = t?2n h(t2 ) = h(t2 ) ; (1 ? t2 )m (t?2 ? 1)n (1 ? t2 )n which is equivalent to the rst identity from (7.10). The second identity follows from Lemma 3.8. The formula from the above theorem can be used to express the face vector of a simplicial complex in terms of the bigraded Betti numbers of the corresponding moment-angle complex ZK . Corollary 7.16. The Euler characteristic of ZK is zero. Proof. We have (ZK ) =
m X

so the statement follows from (7.10). Remark. Another proof of the above corollary follows from the observation that the diagonal subgroup S 1 T m always acts freely on ZK (see section 7.5). Hence, there exists a principal S 1 -bundle ZK ! ZK =S 1 , which implies (ZK ) = 0. The torus Z? = ?1 (1; : : : ; 1) = T m is a cellular subcomplex of ZK , see Lemma 6.18. The cellular cochain subcomplex C (Z? ) C (ZK ) = A (K ) has the basis consisting of cochains T (?; ) and is mapped to the exterior algebra u1; : : : ; um] A (K ) under the isomorphism of Theorem 7.11. It follows that there is an isomorphism of k-modules (7.13) C (ZK ; Z? ) = A (K )= u1; : : : ; um ]: We introduce relative bigraded Betti numbers (7.14) b?q;2p (ZK ; Z? ) = dim H ?q;2p C (ZK ; Z? ); d ; q; p = 0; : : : ; m; de ne the p-th relative Euler characteristic p (ZK ; Z? ) by (7.15)
p (ZK

p;q=0

(?1)?q+2p b?q;2p (ZK ) =

m X p=0

p (ZK ) = (ZK ; 1)

;

; Z? ) =

m X q=0

(?1)q dim C ?q;2p (ZK ; Z? ) =
m X p=0 p (ZK

m X q=0

(?1)q b?q;2p (ZK ; Z? );

and de ne the corresponding generating polynomial: (ZK ; Z? ; t) =

; Z? )t2p :


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7. COHOMOLOGY OF MOMENT-ANGLE COMPLEXES

Theorem 7.17. For any (n ? 1)-dimensional simplicial complex vertices it holds that (7.16) (ZK ; Z? ; t) = (1 ? t2 )m?n (h0 + h1 t2 + + hn t2n ) ? (1 ? Proof. Since C (Z? ) = u1 ; : : : ; um ] and bideg ui = (?1; 2), we ? dim C ?q (Z? ) = dim C ?q;2q (Z? ) = m : q Combining (7.13), (7.9) and (7.15), we get p (ZK ; Z? ) = p (ZK ) ? (?1)p dim C ?p;2p (Z? ): Hence,

K with m t2 )m :
have

(ZK ; Z? ; t) = (ZK ; t) ? = (1 ? t

m X p=0

? (?1)p m t2p p

by (7.10). We will use the above theorem in section 7.6.

2 )m?n

(h0 + h1 t2 +

+ hn t2n ) ? (1 ? t2 )m ;

interpretations of combinatorial results and problems from chapters 2 and 3. Theorem 7.18. Let K be an (n ? 1)-dimensional simplicial sphere, and ZK the corresponding moment-angle manifold, dim ZK = m + n. Then the fundamental cohomology class of ZK is represented by any monomial v u 2 A (K ) of bidegree (?(m ? n); 2m) such that is an (n ? 1)-simplex of K and \ = ?. The sign depends on a choice of orientation for ZK . Proof. Lemma 7.14 (f ) shows that H m+n (ZK ) = H ?(m?n);2m (ZK ). The module A?(m?n);2m (K ) is spanned by the monomials v u such that 2 K n?1, j j = n, = m] n . Every such monomial is a cocycle. Suppose that ; 0 are two (n ? 1)-simplices of K n?1 sharing a common (n ? 2)-face. We claim that the corresponding cocycles v u , v 0 u 0 (where = m] n , 0 = m] n 0 ) represent the same cohomology class up to a sign. Indeed, let v u = vi1 vin uj1 ujm?n ; v 0 u 0 = vi1 vin?1 vj1 uin uj2 ujm?n : Since every (n ? 2)-face of K is contained in exactly two (n ? 1)-faces, the identity d(vi1 vin?1 uin uj1 uj2 ujm?n ) = vi1 vin uj1 ujm?n ? vi1 vin?1 vj1 uin uj2 ujm?n holds in A (K ) u1; : : : ; um] k(K ). Hence, v u ] = v 0 u 0 ] (as cohomology n?1 is a simplicial sphere, every two (n ? 1)-simplices can be classes). Since K connected by a chain of simplices in such a way that any two successive simplices share a common (n?2)-face. Thus, all monomials v u in A?(m?n);2m(K ) represent the same cohomology class (up to a sign). This class is a generator of H m+n (ZK ), i.e. the fundamental cohomology class of ZK .

7.4. Bigraded Betti numbers of ZK : the case of spherical K If K is a simplicial sphere then ZK is a manifold (Lemma 6.13). This imposes additional conditions on the cohomology of ZK and leads to some interesting


7.4. BIGRADED BETTI NUMBERS OF ZK : THE CASE OF SPHERICAL K

111

Simplicial complexes satisfying these two conditions are called pseudomanifolds . In particular, every triangulated manifold is a pseudomanifold. Hence, for any triangulated manifold K n?1 we have bm+n(ZK ) = b?(m?n);2m (ZK ) = 1, and the generator of H m+n(ZK ) can be chosen as described in Theorem 7.18. Corollary 7.19. The Poincare duality for the moment angle manifold ZK corresponding to a simplicial sphere K n?1 respects the bigraded structure in the (co)homology, i.e. H ?q;2p (ZK ) = H?(m?n)+q;2(m?p) (ZK ): In particular, (7.17) b?q;2p (ZK ) = b?(m?n)+q;2(m?p) (ZK ): Corollary 7.20. Let K n?1 be an (n ? 1)-dimensional simplicial sphere, and ZK the corresponding moment-angle complex, dim ZK = m + n. Then (a) b?q;2p (ZK ) = 0 for q > m ? n, with only exception b?(m?n);2m = 1; (b) b?q;2p (ZK ) = 0 for p ? q > n, with only exception b?(m?n);2m = 1. It follows that if K n?1 is a simplicial sphere, then non-zero bigraded Betti numbers br;2p (ZK ) with r 6= 0 and r 6= m ? n appear only in the strip bounded by the lines r = ?(m ? n ? 1), r = ?1, p + r = 1 and p + r = n ? 1 in the second quadrant, see Figure 7.1 (b). Compare this with Figure 7.1 (a) corresponding to the case of general K . Example 7.21. Let K = @ m?1 . Then k(K ) = k v1 ; : : : ; vm ]=(v1 vm ), see Example 3.9. A direct calculation shows that the cohomology H k(K ) u1; : : : ; um]; d] (see Theorem 7.7) is additively generated by the classes 1 and v1 v2 vm?1 um ]. We have deg(v1 v2 vm?1 um ) = 2m ? 1, and Theorem 7.18 says that v1 v2 vm?1 um represents the fundamental cohomology class of ZK = S 2m?1 . Example 7.22. Let K be the boundary complex of an m-gon P 2 with m > 4. We have k(K ) = k v1 ; : : : ; vm ]=IP , where IP is generated by the monomials vi vj , i ? j 6= 0; 1 mod m. The complex ZK = ZP is a manifold of dimension m + 2. The Betti numbers of these manifolds were calculated in 31]. Namely, 8 > 1 for k = 0; m + 2; < k (ZP ) = 0 for k = 1; 2; m; m + 1; (7.18) dim H > ?? ?? ?? : (m ? 2) m?22 ? m?12 ? m?32 for 3 6 k 6 m ? 1: k k k 3 (ZP ) has 5 generators represented For example, in the case m = 5 the group H by the cocycles vi ui+2 2 k(K ) u1; : : : ; u5], i = 1; : : : ; 5, while the group H 4 (ZP ) has 5 generators represented by the cocycles vj uj+2 uj+3 , j = 1; : : : ; 5. As it follows from Theorem 7.18, the product of cocycles vi ui+2 and vj uj+2 uj+3 represents a non-zero cohomology class in H 7 (ZP ) if and only if all the indices i; i + 2; j; j + 2; j + 3 are di erent. Thus, for each of the 5 cohomology classes vi ui+2 ] there is a unique (Poincare dual) cohomology class vj uj+2 uj+3 ] such that the product vi ui+2 ] vj uj+2 uj+3 ] is non-zero. This observation has the following

K n?1. The rst one is that every (n ? 2)-face is contained in exactly two (n ? 1)faces, and the second is that every two (n ? 1)-simplices can be connected by a chain of simplices with any two successive simplices sharing a common (n ? 2)-face.

Remark. In the above proof we have used two combinatorial properties of


112

7. COHOMOLOGY OF MOMENT-ANGLE COMPLEXES

ZP 2 for any m-gon P 2 .

generalization, which describes the multiplicative structure in the cohomology of
Proposition 7.23 (Cohomology ring of ZP 2 ). Let P 2 be an m-gon, m > 4. (a) The only non-zero bigraded cohomology groups of ZP 2 are H 0;0(= H 0 ), H ?p;2(p+1) (= H p+2 ) for p = 1; : : : ; m ? 3, and H ?m+2;2m (= H m+2 ). (b) The group H ?p;2(p+1) is free and is generated by the cohomology classes vi u ] such that j j = p, i 2 and i 1 2 . These cohomology classes are = = subject to relations of the form du 0 = 0 for j 0 j = p + 1. The corresponding

Betti numbers are given by (7.18). (c) The group H ?m+2;2m is one-dimensional with generator v1 v2 u3 um]. (d) The product of two cohomology classes vi1 u 1 ] 2 H ?p1 ;2(p1 +1) and vi2 u 2 ] 2 H ?p2 ;2(p2 +1) equals v1 v2 u3 um] (up to a sign) if ffi1g; fi2g; 1 ; 2 g is a partition of m], and zero otherwise. Therefore, the only non-trivial products in the ring H (ZP 2 ) are those which give a multiple of the fundamental class. Proof. Statement (a) follows from Corollary 7.20, (b) is obvious, and (c) follows from Theorem 7.18. In order to prove (d) we mention that the product of two classes a1 2 H ?p1 ;2(p1 +1) and a2 2 H ?p2 ;2(p2 +1) has bidegree (?p; 2q) with q ?p = 2, whence it can be non-zero only if it belongs to H ?m+2;2m , by Corollary 7.20. It follows from (7.9) and (7.17) that for any simplicial sphere K the following holds: p (ZK ) = (?1)m?n m?p (ZK ): From this and (7.10) we get h0 + h1 t2 + + hn t2n = (?1)m?n m + m?1 t2 + + 0 t2m (1 ? t2 )n (1 ? t2 )m ?2 + ?2m ?2 + ?2n = (?1)n 0 + 1 t(1 ? t?2 )+ m t = (?1)n h0 + h1 t(1 ? t?2 )+ hn t m n 2n 2(n?1) = h0 t + h1 t ? t2 )n+ + hn : (1 Hence, hi = hn?i . Thus, the Dehn{Sommerville equations are a corollary of the bigraded Poincare duality (7.17). The identity (7.10) also allows us to interpret di erent inequalities for the f vectors of simplicial spheres or triangulated manifolds in terms of topological invariants (the bigraded Betti numbers) of the corresponding moment-angle manifolds (or complexes) ZK . Example 7.24. Using the expansion 1 1 m?n = X m ? n + i ? 1 t2i 1 ? t2 i i=0 together with the UBC for simplicial spheres (Corollary 3.19) and identity (7.10), we deduce that the inequality (ZK ; t) 6 1 holds coe cient-wise for any simplicial sphere K n?1. That is, i (ZK ; t) 6 0 for i > 0:


7.5. PARTIAL QUOTIENTS OF ZP

113

(ZK ) = 1; 1 (ZK ) = 0; (ZK ) = ?b?1;4 (ZK ) = ?b3(ZK ); 3 (ZK ) = b?2;6 (ZK ) ? b?1;6 (ZK ) (note that b4 (ZK ) = b?2;6 (ZK ) and b5(ZK ) = b?1;6 (ZK ) + b?3;8 (ZK )). Now, identity (7.10) shows that h0 = 1; h1 = m ? n; ? h2 = m?2n+1 ? b3 (ZK ); ? h3 = m?3n+2 ? (m ? n)b?1;4 (ZK ) + b?2;6 (ZK ) ? b?1;6 (ZK ): It follows that the inequality h1 6 h2 (n > 4) from the GLBC (1.14) for simplicial spheres is equivalent to the following: ? (7.19) b3 (ZK ) 6 m?n : 2 (Note that this inequality is not valid for n = 2, see e.g. Example 7.22, and becomes identity for n = 3.) The next inequality h2 6 h3 (n > 6) from (1.14) is equivalent to the following: ?m?n+1 (7.20) ? (m ? n ? 1)b?1;4 (ZK ) + b?2;6(ZK ) ? b?1;6(ZK ) > 0: 3 We see that the combinatorial GLBC inequalities are interpreted as \topological" inequalities for the (bigraded) Betti numbers of a manifold. This might open a possibility to use topological methods (such as the equivariant topology or Morse theory) for proving inequalities like (7.19) or (7.20). Such a topological approach to problems like g-conjecture or GLBC has an advantage of being independent on whether the simplicial sphere K is polytopal or not. Indeed, as we have already mentioned, all known proofs for the necessity condition in the g-theorem for simplicial polytopes (including the original one by Stanley given in section 5.1, McMullen's proof 97], and the recent proof by Timorin 133]) follow the same scheme. Namely, the numbers hi , i = 1; : : : ; n, are interpreted as the dimensions of graded components Ai of a certain algebra A satisfying the Hard Lefschetz Theorem. The latter means that there is an element ! 2 A1 such that the multiplication by ! denes a monomorphism Ai ! Ai+1 for i < n . This implies hi 6 hi+1 for i < n 2 2 (see section 5.1). However, such an element ! is lacking for non-polytopal K , which means that a new technique has to be developed in order to prove the g-conjecture for simplicial spheres. As it was mentioned in section 3.5, simplicial spheres are Gorenstein* complexes. Using Theorems 3.38, 3.39 and our Theorem 7.6 we obtain the following answer to a weaker version of Problem 6.14. Proposition 7.26. The complex ZK is a Poincare duality complex (over k) if and only if K is Gorenstein*, i.e., for any simplex 2 K (including = ?) the subcomplex link has the homology of a sphere of dimension dim (link ).
0 2

Example 7.25. Using Lemma 7.14 we calculate

Here we return to the case of polytopal K (i.e. K = KP for some simple polytope P ) and study quotients of ZP by freely acting subgroups H T m .

7.5. Partial quotients of Z

P


114

7. COHOMOLOGY OF MOMENT-ANGLE COMPLEXES

For any combinatorial simple polytope P n , de ne s = s(P n ) to be the maximal dimension for which there exists a subgroup H = T s in T m acting freely on ZP . The number s(P n ) is obviously a combinatorial invariant of P n . Problem 7.27 (V. M. Buchstaber). Provide an e cient way to calculate the number s(P n ), e.g. in terms of known combinatorial invariants of P n . Proposition 7.28. If P n has m facets, then s(P n ) 6 m ? n. Proof. Every subtorus of T m of dimension > m ? n intersects non-trivially with any n-dimensional isotropy subgroup, and therefore cannot act freely on ZP .
Proposition 7.29. The diagonal circle subgroup Sd := f(e2 i'; : : : ; e2 i' ) 2 T ' 2 R, acts freely on any ZP . Thus, s(P n ) > 1. Proof. By De nition 6.1, every isotropy subgroup for ZP is coordinate, and therefore intersects Sd only at the unit. An alternative lower bound for the number s(P n ) was proposed in 79]. Let F = fF1 ; : : : ; Fm g be the set of facets of P n . We generalize the de nition of a
m g,

regular coloring from section 6.6 as follows. A surjective map % : F ! k] (where k] = f1; : : : ; kg) is called a regular k-paint coloring of P n if %(Fi ) 6= %(Fj ) whenever Fi \ Fj 6= ?. The chromatic number (P n ) is the minimal k for which there exists a regular k-paint coloring of P n . Then (P n ) > n and, due to the result mentioned in section 6.6, the equality is achieved if and only if every 2-face of P n is an evengon. Note also that (P 3 ) 6 4 by the Four Color Theorem. Example 7.30. Suppose P n is a 2-neighborly simple polytope with m facets. Then (P n ) = m. Proposition 7.31 ( 79]). The fol lowing inequality holds: s(P n ) > m ? (P n ): Proof. The map % : F ! k] de nes an epimorphism of tori % : T m ! T k . ~ It is easy to see that if % is a regular coloring, then Ker % = T m?k acts freely ~ on ZP . For more results on colorings and their relations with Problem 7.27 see 81]. Let H T m be a subgroup of dimension r 6 m ? n. Choosing a basis, we can write it in the form (7.21) H = (e2 i(s11 '1 + +s1r 'r ) ; : : : ; e2 i(sm1 '1 + +smr 'r ) ) 2 T m ; where 'i 2 R, i = 1; : : : ; r. The integer m r-matrix S = (sij ) de nes a monomorphism Zr ! Zm whose image is a direct summand in Zm. For any subset fi1; : : : ; in g m] denote by S^1 ;::: ;^n the (m ? n) r submatrix of S obtained ii by deleting the rows i1 ; : : : ; in. Write each vertex v 2 P n as an intersection of n facets, as in (5.10). The following criterion of freeness for the action of H on ZP holds. Lemma 7.32. Subgroup (7.21) acts freely on ZP if and only if for every vertex v = Fi1 \ : : : \ Fin of P n the (m ? n) r-submatrix S^1 ;::: ;^n de nes a monomorphism ii Zr ,! Zm?n to a direct summand.


7.5. PARTIAL QUOTIENTS OF ZP

115

In particular, for subgroups of rank m ? n we get the following statement. Corollary 7.33. The subgroup (7.21) of rank r = m ? n acts freely on ZP if and only if for any vertex v = Fi1 \ : : : \ Fin of P n holds det S^1 :::^n = 1. ii Proposition 7.34. A simple polytope P n admits a characteristic map if and only if s(P n ) = m ? n. Proof. Proposition 6.5 shows that if P n admits a characteristic map `, then the (m ? n)-dimensional subgroup H (`) acts freely on ZP , whence s(P n ) = m ? n. Now suppose s(P n ) = m ? n, i.e. there exists a subgroup (7.21) of rank r = m ? n that acts freely on ZP . The corresponding m (m ? n)-matrix S de nes a monomorphism Zm?n ! Zm whose image is a direct summand. It follows that there is an n m-matrix such that the sequence 0 ? ?? Zm?n ? ?? Zm ? ?? Zn ? ?? 0 ?! ?S ! ?! ?! is exact. Since S satis es the condition of Corollary 7.33, the matrix satis es (5.5), thus de ning a characteristic map for P n . Suppose M 2n is a quasitoric manifold over P n with characteristic map `. Write the subgroup H (`) in the form (7.21). Now de ne the following linear forms in k v1 ; : : : ; vm]: (7.22) wi = s1i v1 + + smi vm ; i = 1; : : : ; m ? n: Under these assumptions the following statement holds. Lemma 7.35. There is the fol lowing isomorphism of algebras: ? H (ZP ) = Tork w1 ;::: ;wm?n ] H (M 2n ); k ; where the k w1 ; : : : ; wm?n ]-module structure in H (M 2n ) = k v1 ; : : : ; vm ]=IP +J` is de ned by (7.22). Proof. By Theorem 7.13, ? H (ZK ) = Tork v1 ;::: ;vm ]=J` k(K )=J` ; k : The quotient k v1 ; : : : ; vm ]=J` is identi ed with k w1 ; : : : ; wm?n ]. Theorem 7.36. The Leray{Serre spectral sequence of the T m?n-bund le ZP ! 2n col lapses at the E3 term. Furthermore, the fol lowing isomorphism of algebras M holds: ? H (ZP ) = H u1; : : : ; um?n] k(P )=J` ; d ;

Proof. It follows from De nition 6.1 that the orbits of T m-action on ZP corresponding to the vertices of P n have maximal (rank n) isotropy subgroups. The isotropy subgroup corresponding to a vertex v = Fi1 \ : : : \ Fin is the coordinate subtorus Tin;::: ;in T m. Subgroup (7.21) acts freely on ZP if and only if it inter1 sects each isotropy subgroup only at the unit. This is equivalent to the condition that the map H Tin;::: ;in ! T m is injective for any v = Fi1 \ : : : \ Fin . This 1 map is given by the integer m (n + r)-matrix obtained by adding n columns (0; : : : ; 0; 1; 0; : : : ; 0)t (with 1 at the place ij , j = 1; : : : ; n) to S . The map is injective if and only if this enlarged matrix de nes a direct summand in Zm. The latter holds if and only if each S^1 ;::: ;^n de nes a direct summand. ii


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7. COHOMOLOGY OF MOMENT-ANGLE COMPLEXES

where

bideg vi = (0; 2); bideg ui = (?1; 2); d(ui ) = wi ; d(vi ) = 0: Proof. Since H (T m?n ) = u1; : : : ; um?n] and H (M 2n ) = k(P )=J` , we have E3 = H (k(P )=J` ) u1 ; : : : ; um?n ]; d : By Lemma 3.29, ? H (k(P )=J` ) u1; : : : ; um?n]; d = Tork w1 ;::: ;wm?n] H (M 2n ); k : Combining the above two identities with Lemma 7.35 we get E3 = H (ZP ), which concludes the proof. Our next aim is to calculate the cohomology of the quotient ZP =H for arbitrary freely acting subgroup H . First, we write H in the form (7.21) and choose an (m ? r) m-matrix T = (tij ) of rank (m ? r) satisfying T S = 0. This is done in the same way as in the proof of Proposition 7.34. In particular, if r = m ? n then T is the characteristic matrix for the quasitoric manifold ZP =H . Theorem 7.37. The fol lowing isomorphism of algebras holds: ? H (ZP =H ) = Tork t1 ;::: ;tm?r ] k(P ); k ; where the k t1 ; : : : ; tm?r ]-module structure on k(P ) = k v1 ; : : : ; vm ]=IP is given by the map k t1 ; : : : ; tm?r ] ! k v1 ; : : : ; vm ] ti ! ti1 v1 + + tim vm : Remark. Theorem 7.37 reduces to Theorem 7.6 in the case r = 0 and to Example 7.5 in the case r = m ? n. Proof of Theorem 7.37. The inclusion T r = H ,! T m de nes the map h : B T r ! B T m of the classifying spaces. Let us consider the commutative square

E ? ?? BT P ?! ? ?
? y ? y

p

where the left vertical arrow is the pullback along h. The space E is homotopy equivalent to the quotient ZP =H . Hence, the Eilenberg{Moore spectral sequence of the above square converges to the cohomology of ZP =H . Its E2 -term is ? E2 = Tork v1 ;::: ;vm ] k(P ); k w1 ; : : : ; wr ] ; where the k v1 ; : : : ; vm ]-module structure in k w1 ; : : : ; wr ] is de ned by the matrix S , i.e. by the map vi ! si1 w1 + : : : + sir wr . In the same way as in the proof of Theorem 7.6 we show that the spectral sequence collapses at the E2 term and the following isomorphism of algebras holds: ? (7.23) H (ZP =H ) = Tork v1 ;::: ;vm ] k(P ); k w1 ; : : : ; wr ] :

?h ! B T r ? ?? B T m ;


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117

Now put = k v1 ; : : : ; vm ], ? = k t1 ; : : : ; tm?r ], A = k w1 ; : : : ; wr ] and C = k(P ) in Theorem 3.36. Since is a free ?-module and = ==? = k w1 ; : : : ; wr ], a ee spectral sequence fEs ; ds g arises. Its E2 term is ? e E2 = Tork w1 ;::: ;wr ] k w1 ; : : : ; wr ]; Tork t1 ;::: ;tm?r ] k(P ); k ; and it converges to Tork v1 ;::: ;vm ] (k(P ); k w1 ; : : : ; wr ]). Obviously, k w1 ; : : : ; wr ] is a free k w1 ; : : : ; wr ]-module, so we have ? e p;q e0 E2 = 0 for p 6= 0; E2 ; = Tork t1 ;::: ;tm?r ] k(P ); k : Thus, the spectral sequence collapses at the E2 term, and the following isomorphism of algebras holds: ? ? Tork v1 ;::: ;vm ] k(P ); k w1 ; : : : ; wr ] = Tork t1;::: ;tm?r ] k(P ); k ; which together with (7.23) concludes the proof. Corollary 7.38. H (ZP =H ) = H u1 ; : : : ; um?r ] k(P n ); d , where dui = (ti1 v1 + : : : + tim vm ), dvi = 0, bideg vi = (0; 2), bideg ui = (?1; 2). Example 7.39. Let H = Sd is the diagonal subgroup. Then the matrix S is a column of m units. By Theorem 7.37, ? (7.24) H (ZP =Sd) = Tork t1 ;::: ;tm?1 ] k(P ); k ; where the k t1 ; : : : ; tm?1 ]-module structure in k(P ) = k v1 ; : : : ; vm ]=I is de ned by ti ?! vi ? vm ; i = 1; : : : ; m ? 1: 1 -bundle ZP ! ZP =Sd is classi ed by a map c : ZP =Sd ! Suppose that the S B T 1 = C P 1 . Since H (C P 1 ) = k w], the element c (w) 2 H 2 (ZP =Sd) is de ned. Lemma 7.40. P n is q-neighborly if and only if (c (w))q 6= 0. Proof. The map c takes the cohomology ring H (B T 1 ) = k w] to the subalgebra ? 0 k(P ) k t1;::: ;tm?1] k = Tork t1;::: ;tm?1] k(P ); k H (ZP =H ): This subalgebra is isomorphic to the quotient k(P )=(v1 = = vm ). Now the assertion follows from the fact that a polytope P n is q-neighborly if and only if the ideal IP does not contain monomials of degree < q + 1. Here we assume that K n?1 is a triangulated manifold. In this case the corresponding moment-angle complex ZK is not a manifold in general, however, its singularities can be easily treated. Indeed, the cubical complex cc(K ) (Construction 4.9) is homeomorphic to j cone(K )j and the vertex of the cone is p = (1; : : : ; 1) 2 cc(K ) I m . Let U" (p) cc(K ) be a small neighborhood of p in cc(K ). The closure of U" (p) is also homeomorphic to j cone(K )j. It follows from the de nition of ZK (see (6.3)) that U" (Z? ) := ?1 (U" (p)) ZK is a small invariant neighborhood of the torus Z? = ?1 (p) = T m in ZK . For small " the closure of U" (Z? ) is homeomorphic to j cone(K )j T m. Removing U" (Z? ) from ZK we obtain a manifold with boundary, which we denote WK . Thus, we have WK = ZK n U" (Z? ); @ WK = jK j T m: Note that since U" (Z? ) is a T m-stable subset, the torus T m acts on WK .

7.6. Bigraded Poincare duality and Dehn{Sommerville equations


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7. COHOMOLOGY OF MOMENT-ANGLE COMPLEXES

Theorem 7.41. The manifold (with boundary) WK is equivariantly homotopy equivalent to the moment-angle complex WK (see (6.3)). Also, there is a canonical relative homeomorphism of pairs (WK ; @ WK ) ! (ZK ; Z? ). Proof. To prove the rst assertion we construct homotopy equivalence cc(K )n U" (p) ! cub(K ) as it is shown on Figure 7.2. This map is covered by an equivariant homotopy equivalence WK = ZK n U" (Z? ) ! WK . The second assertion follows easily from the de nition of WK .
? ? J Xy ??X X J ? ? XX X] ? J ? ? J @ R J C@ @ J C + CC W C? C ??? ? ? ? ?

cub(K )

u u

u

u

u u

cc(K ) n U" (p)

Figure 7.2. Homotopy equivalence cc(K ) n U" (p)

! cub(K ).

By Lemma 6.15, the moment-angle complex WK (D2 )m has a cellular structure with 5 di erent cell types Di , Ii , 0i , Ti , 1i , i = 1; : : : ; m (see Figure 6.1). The homology of WK (and therefore of WK ) can be calculated from the corresponding cellular chain complex, which we denote C (WK ); @c ]. Although WK has more types of cells than ZK (5 instead of 3), its cellular chain complex C (WK ); @c ] also has a natural bi grading. Namely, the following statement holds (compare with (7.4)). Lemma 7.42. Put (7.25) bideg Di = (0; 2); bideg Ti = (?1; 2); bideg Ii = (1; 0); bideg 0i = bideg 1i = (0; 0); i = 1; : : : ; m: This turns the cel lular chain complex C (WK ); @c ] into a bigraded di erential module with di erential @c adding (?1; 0) to bidegree. The original grading of C (WK ) by the dimension of cel ls corresponds to the total degree (i.e. the dimension of a cel l equals the sum of its two degrees). Proof. The only thing we need to check is that the di erential @c adds (?1; 0) to bidegree. This follows from (7.25) and the following formulae: @c Di = Ti ; @c Ii = 1i ? 0i ; @c Ti = @c 1i = @c 0i = 0: Unlike the bigraded rst degree (due to change the second bigraded complex C structure in C (ZK ), elements of C ; (WK ) may the positive rst degree of Ii ). The di erential degree (as in the case of ZK ), which allows us ; (WK ) into the sum of complexes C ;2p (WK ), p have positive @c does not to split the = 0; : : : ; m.


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119

In the same way as we did this for ZK and (ZK ; Z? ) we de ne (7.26) bq;2p (WK ) = dim Hq;2p C ; (WK ); @c ; ?m 6 q 6 m; 0 6 p 6 m; (7.27)
p (WK ) = m X q=?m

(?1)q dim Cq;2p (WK ) =
m X p=0

m X

q=?m

(?1)q bq;2p (WK );

(WK ; t) =

p (WK )t2p ;

(note that q above may be negative). The following theorem gives a formula for the generating polynomial (WK ; t) and is analogous to theorems 7.15 and 7.17. Theorem 7.43. For any simplicial complex K n?1 with m vertices it holds that ? (WK ; t) = (1 ? t2 )m?n (h0 + h1 t2 + + hn t2n ) + (K ) ? 1 (1 ? t2 )m = (1 ? t2 )m?n (h0 + h1 t2 + + hn t2n ) + (?1)n?1 hn (1 ? t2 )m ; where (K ) = f0 ? f1 + : : : + (?1)n?1 fn?1 = 1 + (?1)n?1 hn is the Euler characteristic of K . Proof. By the de nition of WK (see (6.3)), the vector R 2 fD; I ; 0; T ; 1gm (see section 6.3) represents a cell of WK if and only if the following two conditions are satis ed: (a) The set RD RI R0 is a simplex of K n?1 . (b) jR0 j > 1. Let cijlpq (WK ) denote the number of cells R WK with jRD j = i, jRI j = j , jR0 j = l, jRT j = p, jR1 j = q, i + j + l + p + q = m. It follows that ? ? ? ? cijlpq (WK ) = fi+j+l?1 i+ji +l j+l m?ip j?l ; (7.28) l where (f0 ; : : : ; fn?1 ) is the f -vector of K (we also assume f?1 = 1 and fk = 0 for k < ?1 or k > n ? 1). By (7.25), ? ? bideg R = jRI j ? jRT j; 2(jRD j + jRT j) = j ? p; 2(i + p) : Now we calculate r (WK ) using (7.27) and (7.28): X ? ? ? ? (?1)j?p fi+j+l?1 i+j+l j+l m?ip j?l : r (WK ) = i l Substituting s = i + j + l above we obtain X ?s ? ?? (?1)s?r?l fs?1 r?p s?r+p mp s r (WK ) = l
l;s;p l>1
X

i;j;l;p i+p=r;l>1

= Since

s;p

?s ? ? (?1)s?r fs?1 r?p mp s

X

l>1

? (?1)l s?r+p l

X

?1; s > r ? p; ? (?1)l s?r+p = 0; s 6 r ? p; l l>1


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7. COHOMOLOGY OF MOMENT-ANGLE COMPLEXES
X ?s ? ? (?1)s?r fs?1 r?p mp s X ?? (?1)r?sfs?1 m?ss : r

we get
r (WK ) =

?

=?

s;p s>r?p
X

s;p

?s ? ? (?1)r?sfs?1 r?p mp s +

s

The second sum in the above formula is exactly r (ZK ) (see (7.11)). To calculate the rst sum we observe that
X?

p

?m s ?m?s r?p p = r

:

This follows from calculating the coe cient of r in the two sides of the identity (1 + )s (1 + )m?s = (1 + )m . Hence, (K ) ? 1 + r (ZK ) s P since ? s (?1)s fs?1 = (K ) ? 1. Finally, using (7.10), we calculate m m m X X X ?? (WK ; t) = (WK )t2r = (?1)r m (K ) ? 1 t2r + r r (ZK )t2r r r? =0 r=0 r=0 2 )m + (1 ? t2 )m?n (h + h t2 + + h t2n ): = (K ) ? 1 (1 ? t 0 1 n
r (WK ) =

?

X

? ? (?1)r?sfs?1 m + r (ZK ) = (?1)r m r r

?

;

Suppose that K is an orientable triangulated manifold. It is easy to see that then WK is also orientable. Hence, there are relative Poincare duality isomorphisms: (7.29)

Hk (WK ) = H m+n?k (WK ; @ WK ); k = 0; : : : ; m:

Theorem 7.44 (Dehn{Sommerville equations for triangulated manifolds). The fol lowing relations hold for the h-vector (h0 ; h1 ; : : : ; hn ) of any triangulated manifold K n?1 :

seen in the same way as in Corollary 7.19 that relative Poincare duality isomorphisms (7.29) regard the bigraded structures in the (co)homology of WK and (ZK ; Z? ). Hence,

? ? hn?i ? hi = (?1)i (K n?1 ) ? (S n?1 ) n ; i = 0; 1; : : : ; n; i where (S n?1 ) = 1 + (?1)n?1 is the Euler characteristic of an (n ? 1)-sphere. Proof. Suppose rst that K is orientable. By Theorem 7.41, Hk (WK ) = Hk (WK ) and H m+n?k (WK ; @c WK ) = H m+n?k (ZK ; Z? ). Moreover, it can be

(7.30)

b?q;2p (WK ) = b?(m?n)+q;2(m?p)(ZK ; Z? ); p (WK ) = (?1)m?n m?p (ZK ; Z? ); (WK ; t) = (?1)m?n t2m (ZK ; Z? ; 1 ): t


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121

Using (7.16), we calculate (?1)m?nt2m (ZK ; Z? ; 1 ) t m?n t2m (1 ? t?2 )m?n (h0 + h1 t?2 + + hn t?2n ) = (?1) ? (?1)m?n t2m (1 ? t?2 )m = (1 ? t2 )m?n (h0 t2n + h1 t2n?2 + + hn ) + (?1)n?1(1 ? t2 )m : Substituting the formula for (WK ; t) from Theorem 7.43 and the above expression into (7.30) we obtain ? (1 ? t2 )m?n (h0 + h1 t2 + + hn t2n ) + (K ) ? 1 (1 ? t2 )m = (1 ? t2 )m?n (h0 t2n + h1 t2n?2 + + hn ) + (?1)n?1(1 ? t2 )m : Calculating the coe cient of t2i in both sides after dividing ?the above identity by (1 ? t2 )m?n , we get hn?i ? hi = (?1)i ( (K n?1 ) ? (S n?1 )) n , as required. i Now suppose that K is non-orientable. Then there exist an orientable triangulated manifold L of the same dimension and a 2-sheet covering L ! K . Then we obviously have fi (L) = 2fi (K ), i = 0; 1; : : : ; n ? 1. It follows from (1.7) that
n X i=0

h

i (L)tn?i

? (t ?

1)n = 2

n X i=0

hi (K )tn?i ? (t ? 1)n :

Hence,

? hi (L) = 2hi (K ) ? (?1)i n ; i = 0; 1; : : : ; n: i ? Since L is orientable, we have hn?i (L) ? hi (L) = (?1)i ( (L) ? (S n?1 )) n . Therei

fore, ? 2 hn?i (K ) ? Since (L) = 2 ? 2 hn?i (K ) ?

as required. If jK j = S n?1 or n ? 1 is odd then Corollary 7.44 gives the classical equations hn?i = hi . Corollary 7.45. Suppose K n?1 is a triangulated manifold with the h-vector (h0 ; : : : ; hn ). Then ? hn?i ? hi = (?1)i (hn ? 1) n ; i = 0; 1; : : : ; n: i Proof. Since (K n?1 ) = 1 + (?1)n?1hn and (S n?1 ) = 1 + (?1)n?1 , we have (K n?1 ) ? (S n?1 ) = (?1)n?1(hn ? 1) = (hn ? 1) (the coe cient (?1)n?1 can be dropped since for odd n ? 1 the left hand side is zero). Corollary 7.46. For any (n ? 1)-dimensional triangulated manifold the numbers hn?i ? hi , i = 0; 1; : : : ; n, are homotopy invariants. In particular, they do not depend on a triangulation.

? ? ? ? hi (K ) ? (?1)n?i nn i + (?1)i n = (?1)i (L) ? (S n?1 ) n : ? i i (K ), we get ? hi (K ) = (?1)i 2 (K ) ? (S n?1 ) + (?1)n ? 1) ? = 2 (?1)i (K ) ? (S n?1 ) ;


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7. COHOMOLOGY OF MOMENT-ANGLE COMPLEXES

In the case of P L-manifolds the topological invariance of numbers hn?i ? hi was observed by Pachner in 110, (7.11)].
6? ? ?6 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?6? ? 6 ? ? ? ? ?@ ? ? ? @ ? @? ? ? ? ? ? ?-

(a) f = (9; 27; 18), h = (1; 6; 12; ?1)

(b) f = (7; 21; 14), h = (1; 4; 10; ?1)

Figure 7.3. \Symmetric" and \minimal" triangulation of T 2 Example 7.47 (Triangulations of 2-manifolds). Consider triangulations of the 2-torus T 2 . We have n = 3, (T 2 ) = 0. From (K n?1 ) = 1 + (?1)n?1hn we deduce h3 = ?1. Corollary 7.44 gives h3 ? h0 = ?2; h2 ? h1 = 6: For instance, the triangulation on Figure 7.3 (a) has f0 = 9 vertices, f1 = 27 edges and f2 = 18 triangles. (Note that this triangulation is the canonical triangulation of @ 2 @ 2 , as described in Construction 2.11.) The corresponding h-vector is (1; 6; 12; ?1). On the other hand, it is well known that a triangulation of T 2 with only 7 vertices can be achieved, see Figure 7.3 (b). Note that this triangulation is neighborly, i.e. its 1-skeleton is a complete graph on 7 vertices. It turns out that no triangulation of T 2 with smaller number of vertices exists. Suppose now K 2 is a 2-dimensional triangulated manifold with m vertices. Let = (K 2 ) be its Euler characteristic. Using Corollary 7.44 we may express the f -vector of K 2 via and m, namely, ? f (K 2 ) = m; 3(m ? ); 2(m ? ) : Since the number of edges in a triangulation does not exceed the number of pairs of vertices, we get the inequality (7.31) 6(m ? ) 6 m(m ? 1); from which a lower bound for the number of vertices in a triangulation of K 2 can be deduced. For instance, in the case of torus T 2 we have = 0 and (7.31) gives m > 7. Note that a minimal triangulation of K 2 is neighborly (has a complete graph as its 1-skeleton) only if (7.31) turns to equality. We have seen that this is the case for T 2 ( = 0, m = 7). Other examples are the sphere S 2 ( = 2, m = 4) and the real pro jective plane RP 2 ( = 1, m = 6). A neighborly triangulation of RP 2 is shown in Figure 7.4. However, for most values of there is no m which makes (7.31) an equality. For example, minimal triangulations of orientable surfaces of genus 1 to 5 are not neighborly. A genus 6 surface (having = ?10 and m = 12) has neighbourly triangulations (which are automatically minimal). These triangulations are important in the problem of polyhedral embeddability of


7.6. BIGRADED POINCARE DUALITY AND DEHN{SOMMERVILLE EQUATIONS
*HH HH H j H 6 A A A A A A A A A A A ? Y H A@ HH ? ? H@ HH? @

123

Figure 7.4. Neighborly triangulation of RP 2 , with f = (6; 15; 10).

orientable triangulated surfaces in R3 (\polyhedral" here means with at triangles and no self-intersections). It was shown in 4] that there are in total 59 di erent neighborly triangulations of a genus 6 surface with 12 vertices. Later, using an algorithm for generating oriented matroids, Bokowski and Guedes de Oliveira proved in 23] that one of these triangulations cannot be embedded into R3 with at triangles. Furthermore, they proved that one triangle can be removed from the triangulation while retaining non-embeddability, so an arbitrary number of handles can be attached at this triangle to get a non-embeddable triangulated surface of any genus > 6. A number of results on minimal triangulations were obtained by Lutz in 91] using his computer program BISTELLAR (which we already mentioned in section 2.3).


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7. COHOMOLOGY OF MOMENT-ANGLE COMPLEXES


CHAPTER 8

Cohomology rings of subspace arrangement complements
8.1. General arrangements and their complements Definition 8.1. An arrangement is a nite set A = fL1; : : : ; Lr g of a ne subspaces in some a ne space (either real or complex). An arrangement A is called a subspace arrangement (or central arrangement ) if all its subspaces are linear (i.e., contain 0). Given an arrangement A = fL1; : : : ; Lr g in C m , de ne its support (or union ) jAj as r jAj := Li C m ;
and its complement U (A) as
i=1

and similarly for arrangements in Rm . Let A = fL1; : : : ; Lr g be an arrangement. The intersections v = Li1 \ \ Lik form a poset (L; <) with respect to the inclusion, called the intersection poset of the arrangement. The poset L is assumed to have a unique maximal element T corresponding to the ambient space of the arrangement. The rank function d on L is de ned by d(v) = dim v. The complex ord(L) (see Example 2.17) is called the order complex of arrangement A. De ne intervals L(v;w) = fx 2 L : v < x < wg; L>v = fx 2 L : x > vg: Arrangements and their complements play a pivotal r^ in many construcole tions of combinatorics, algebraic and symplectic geometry etc.; they also arise as con guration spaces for di erent classical mechanical systems. In the study of arrangements it is very important to get a su ciently detailed description of the topology of complements U (A) (this includes number of connected components, homotopy type, homology groups, cohomology ring, etc.). A host of elegant results in this direction appeared during the last three decades, however, the whole picture is far from being complete. The theory ascends to work of Arnold 6], in which the classifying space for the colored braid group is described as the complement of the arrangement of all diagonal hyperplanes fzi = zj g, 1 6 i < j 6 n, in C n . The latter complement can be thought as the con guration space of n ordered points in C . Its cohomology ring was also calculated in 6]. This result was generalized by Brieskorn 29] and motivated the further development of the theory of complex hyperplane arrangements (i.e. arrangements of codimension-one complex a ne subspaces). One of the main results here is the following.
125

U (A) := C m n jAj;


126

8. COHOMOLOGY RINGS OF SUBSPACE ARRANGEMENT COMPLEMENTS

Relations between the forms !j = 21 i dljj , j = 1; : : : ; r, were explicitly del scribed by Orlik and Solomon 107]. We give their result in the central case, i.e. when all the hyperplanes are vector subspaces. Then there is one relation
p X

Theorem 8.2 ( 6], 29]). Let A = fL1; : : : ; Lr g be an arrangement of complex hyperplanes in C m , where the hyperplane Lj is the zero set of linear function lj , j = 1; : : : ; r. Then the integer cohomology algebra of the complement C m n jAj is isomorphic to the algebra generated by closed di erential 1-forms 21 i dljj . l

for any minimal subset fLj1 ; : : : ; Ljp g of hyperplanes of A such that codim Lj1 \ \ Ljp = p ? 1 (such subsets are called circuits of L). Example 8.3. Let A be the arrangement of diagonal hyperplanes fzj = zk g, ? 1 6 j < k 6 n, in C n . Then we have the forms !jk = 21 i d(zzjj?zzkk ) , satisfying the identities !ij ^ !jk + !jk ^ !ki + !ki ^ !ij = 0; known as the Arnold relations . The theory of complex hyperplane arrangements is probably the most well understood part of the whole study. Several surveys and monographs are available; we mention just 108], 136] and 143], where further references can be found. Relationships between real hyperplane arrangements, polytopes and oriented matroids are discussed in 145, Lecture 7]. Another interesting related class of arrangements is known as 2-arrangements in R2n . A 2-arrangement is an arrangement of real subspaces of codimension 2 with even-dimension intersections. In particular, any complex hyperplane arrangement is a 2-arrangement. The relationships between 2-arrangements and complex hyperplane arrangements are studied in 144]. In the case of general arrangement A, the celebrated Goresky{MacPherson theorem 66, Part III] expresses the cohomology groups H i (U (A)) (without ring structure) as a sum of homology groups of subcomplexes of a certain simplicial complex. Theorem 8.4 (Goresky and MacPherson 66, Part III]). The (co)homology of a subspace arrangement complement U (A) in Rn : is given by M ? ? Hi U (A); Z = H n?d(v)?i?1 ord(L>v ); ord(L(v;T ) ); Z ;

k=1

(?1)k !j1 ^

^ !jk ^ c

^ !jp = 0;

H U (A); Z =

i

?

v2P

M

(see De nition 8.1), where we assume The original proof of this theorem in 66]. Remark. Observing that ord(L>v the formula from Theorem 8.4 as M ? e e (8.1) Hi U (A); Z = H

v2P

Hn?d(v)?i?1 ord(L>v ); ord(L(v;T ) ); Z ;
that H?1 (?; ?) = H ?1 (?; ?) = Z. used the strati ed Morse theory , developed

?

) is the cone over ord(L(v;T ) ), we may rewrite
n?d(v)?i?2 ?ord(L(v;T ) );

and similarly for the cohomology.

v2P

Z;


8.2. COORDINATE SUBSPACE ARRANGEMENTS

127

A comprehensive survey of general arrangements is given in 20]. Monograph 136] gives an alternative approach to homology and homotopy computations, via the Anderson spectral sequence . A method of describing the homotopy types of subspace arrangements, using diagrams of spaces over poset categories, was proposed in 146]. Using this method, a new, elementary, proof of Goresky{ MacPherson Theorem 8.4 was found there. The approach of 146] was developed later in 139] by incorporating homotopy colimits techniques. The cohomology rings of arrangement complements are much more subtle. In general, the integer cohomology ring of U (A) is not determined by the intersection poset L (this is false even for 2-arrangements, as shown in 144]). An approach to calculating the cohomology algebra of the complement U (A), based on the results of De Concini and Procesi 51], was proposed by Yuzvinsky in 142]. Recently, the combinatorial description of the product of any two cohomology classes in a complex subspace arrangement complement U (A), conjectured by Yuzvinsky, has been obtained independently in 53] and 55]. This description is given in terms of the intersection poset L(A), the dimension function, and additional orientation data.

Remark. The homology groups of a complex arrangement in C n can be calculated by regarding it as a real arrangement in R2n .

is a coordinate subspace. In this section we apply the results of chapter 7 to cohomology algebras of complex coordinate subspace arrangement complements. The case of real coordinate arrangements is also discussed at the end of this section. A coordinate subspace of C m can be written as (8.2) L = f(z1; : : : ; zm) 2 C m : zi1 = = zik = 0g; where = fi1; : : : ; ik g is a subset of m]. Obviously, dim L = m ? j j. Construction 8.5. For each simplicial complex K on the set m] de ne the complex coordinate subspace arrangement C A(K ) by C A(K ) = fL : 2 K g: = Denote the complement of C A(K ) by U (K ), that is (8.3) U (K ) = C m n L : Note that if K 0 K is a subcomplex, then U (K 0 ) U (K ). Proposition 8.6. The assignment K 7! U (K ) de nes a one-to-one orderpreserving correspondence between the set of simplicial complexes on m] and the set of coordinate subspace arrangement complements in C m (or Rm ). Proof. Suppose C A is a coordinate subspace arrangement in C m . De ne (8.4) K (C A) := f m] : L 6 jC Ajg: Obviously, K (C A) is a simplicial complex. By the de nition, K (C A) depends only on jC Aj (or U (C A)) and U (K (C A)) = U (C A), whence the proposition follows.
2K =

8.2. Coordinate subspace arrangements and the cohomology of ZK . An arrangement A = fL1; : : : ; Lr g is called coordinate if every Li , i = 1; : : : ; r,


128

8. COHOMOLOGY RINGS OF SUBSPACE ARRANGEMENT COMPLEMENTS

If C A contains a hyperplane, say fzi = 0g, then its complement U (C A) is factored as U (C A0 ) C , where C A0 is a coordinate subspace arrangement in the hyperplane fzi = 0g and C = C n f0g. Thus, for any coordinate subspace arrangement C A, the complement U (C A) decomposes as U (C A) = U (C A0 ) (C )k ; were C A0 is a coordinate arrangement in C m?k that does not contain hyperplanes. On the other hand, (8.4) shows that C A contains the hyperplane fzi = 0g if and only if fig is not a vertex of K (C A). It follows that U (K ) is the complement of a coordinate arrangement without hyperplanes if and only if the vertex set of K is the whole m]. Keeping in mind these remarks, we restrict our attention to coordinate subspace arrangements without hyperplanes and simplicial complexes on the vertex set m]. Remark. In the notations of Construction 6.38 we have U (K ) = K (C ; C ). Example 8.7. 1. If K = m?1 then U (K ) = C m . 2. If K = @ m?1 (boundary of simplex) then U (K ) = C m n f0g. 3. If K is a disjoint union of m vertices, then U (K ) is the complement in C m of the set of all codimension-two coordinate subspaces zi = zj = 0, 1 6 i < j 6 m. The diagonal action of algebraic torus (C )m on C m descends to U (K ). In particular, there is the standard action of T m on U (K ). The quotient U (K )=T m m can be identi ed with U (K ) \ R+ , where Rm is regarded as a subset of C m . + m and ZK Lemma 8.8. cc(K ) U (K ) \ R+ U (K ) (see Construction 4.9 and (6.3)). Proof. Take y = (y1 ; : : : ; ym ) 2 cc(K ). Let = fi1 ; : : : ; ik g be the set of zero coordinates of y, i.e. the maximal subset of m] such that y 2 L \ Rn . + Then it follows from the de nition of cc(K ) (see (4.4)) that is a simplex of K . Hence, L 2 C A(K ) and y 2 U (K ), which implies the rst statement. The second = assertion follows from the fact that cc(K ) is the quotient of ZK . Theorem 8.9. There is an equivariant deformation retraction U (K ) ! ZK . Proof. First, we construct a deformation retraction r : U (K ) \ Rm ! cc(K ). + This is done inductively. We start from the boundary complex of an (m ? 1)-simplex and remove simplices of positive dimensions until we obtain K . On each step we construct a deformation retraction, and the composite map will be the required retraction r. If K = @ m?1 is the boundary complex of an (m ? 1)-simplex, then U (K ) \ m = Rm n f0g. In this case the retraction r is shown on Figure 8.1. Now suppose R+ + that K is obtained from K 0 by removing one (k ? 1)-dimensional simplex = fj1 ; : : : ; jk g, that is K = K 0 . By the inductive hypothesis, we may assume that there is a deformation retraction r0 : U (K 0 ) \ Rm ! cc(K 0 ). Let a 2 Rm be the + + point with coordinates yj1 = : : : = yjk = 0 and yi = 1 for i 2 . Since is not = a simplex of K , we have a 2 U (K ) \ Rm . At the same time, a 2 C (see (4.1)). = + Hence, we can apply the retraction shown on Figure 8.1 on the face C I m, 0 is the required with center at a. Denote this retraction by r . Then r = r r deformation retraction. The deformation retraction r : U (K )\Rm ! cc(K ) is covered by an equivariant + deformation retraction U (K ) ! ZK , which concludes the proof.


8.2. COORDINATE SUBSPACE ARRANGEMENTS
? ?

u e

6 ?

? 7? ? ?> ? * : -

?

u u

129

?

?

Figure 8.1. The retraction r : U (K )

\ Rm ! cc(K ) for K = @ +

m?1 .

In the case K = KP (i.e. K is a polytopal simplicial sphere corresponding to a simple polytope P n ) the deformation retraction U (KP ) ! ZP from Theorem 8.9 can be realized as the orbit map for an action of a contractible group. We denote U (P n ) := U (KP ). Set Rm = f(y1 ; : : : ; ym ) 2 Rm : yi > 0; i = 1; : : : ; mg Rm : > + m is a group with respect to the multiplication, and it acts on Rm , C m and Then R> U (P n ) by coordinatewise multiplications. There is the isomorphism exp : Rm ! Rm between the additive and the multiplicative groups taking (y1 ; : : : ; ym ) 2 Rm > to (ey1 ; : : : ; eym ) 2 Rm . > Let us consider the m (m ? n)-matrix W introduced in Construction 1.8 for every simple polytope (1.1). Proposition 8.10. For any vertex v = Fi1 \ \ Fin of P n the maximal minor W^1 :::^n of W obtained by deleting n rows i1 ; : : : ; in is non-degenerate: det W^1 :::^n 6= ii ii 0. Proof. If det W^1 :::^n = 0 then the vectors l i1 ; : : : ; l in (see (1.1)) are linearly ii dependent, which is impossible. The matrix W de nes the subgroup (8.5) RW = (ew11 1 + +w1;m?n m?n ; : : : ; ewm1 1 + +wm;m?n m?n ) Rm ; > m?n . Obviously, RW = Rm?n . where ( 1 ; : : : ; m?n ) is running through R > Theorem 8.11 ( 33, Theorem 2.3] and 38, x3]). The subgroup RW acts freely on U (P n ) C m . The composition ZP ,! U (P n ) ! U (P n )=RW of the embedding ie (Lemma 6.6) and the orbit map is an equivariant di eomorphism (with respect to the corresponding T m-actions). Suppose now that P n is a lattice simple polytope, and let MP be the corresponding toric variety (Construction 5.4). Along with the real subgroup RW Rm (8.5) de ne its complex analogue > CW = (ew11 1 + +w1;m?n m?n ; : : : ; ewm1 1 + +wm;m?n m?n ) (C )m ; where ( 1 ; : : : ; m?n ) is running through C m?n . Obviously, CW = (C )m?n . It is shown in 45] (see also 9], 16]) that CW acts freely on U (P n ) and the toric variety


130

8. COHOMOLOGY RINGS OF SUBSPACE ARRANGEMENT COMPLEMENTS

Thus, we have the following commutative diagram: (8.6)
? y

MP can be identi ed with the orbit space (or geometric quotient ) U (P n )=CW .
RW =Rm?n > U (P n ) ???????! ZP ? ?

CW =(C )m?n

? m?n yT

Remark. It can be shown 45, Theorem 2.1] that any toric variety M corresponding to a fan Rn with m one-dimensional cones can be identi ed with the universal categorical quotient U (C A )=G, where U (C A ) a certain coordinate arrangement complement (determined by the fan ) and G = (C )m?n . The categorical quotient becomes the geometric quotient if and only if the fan is simplicial. In this case U (C A ) = U (K ). On the other hand, if the pro jective toric variety MP is non-singular then MP is a symplectic manifold of dimension 2n, and the action of T n on it is Hamiltonian (see e.g. 9] or 45, x4]). In this case the diagram (8.6) displays MP as the result of a symplectic reduction . Namely, let HW = T m?n be the maximal compact subgroup in CW , and : C m ! Rm?n the moment map for the Hamiltonian action of HW on C m . Then for any regular value a 2 Rm?n of the map there is the following di eomorphism: ?1 (a)=HW ?! U (P n )=CW = MP (details can be found in 9]). In this situation ?1 (a) is exactly our manifold ZP . This gives us another interpretation of the manifold ZP as the level surface for the moment map (in the case when P n can be realized as the quotient of a non-singular pro jective toric variety). Example 8.12. Let P n = n (the n-simplex). Then m = n + 1, U (P n ) = n+1 n f0g. Moreover, RW = R> , CW = C and HW = S 1 are the diagonal C subgroups in Rn+1 , (C )n+1 and T m+1 respectively (see Example 1.9). Hence, > ? ? ZP = S 2n+1 = C n+1 n f0g =R> ; MP = C n+1 n f0g =C = C P n : The moment map : C m ! R takes (z1 ; : : : ; zm) 2 C m to 1 (jz1 j2 + : : : + jzm j2 ), 2 and for a 6= 0 we have ?1 (a) = S 2n+1 = ZK . Now we have the following result for the cohomology of subspace arrangement complements. Theorem 8.13. The fol lowing isomorphism of graded algebras holds: ? ? H U (K ) = Tork v1 ;::: ;vm ] k(K ); k = H u1; : : : ; um] k(K ); d : Proof. This follows from Theorems 8.9, 7.6 and 7.7. Theorem 8.13 provides an e ective way to calculate the cohomology algebra of the complement of any complex coordinate subspace arrangement. The Koszul complex was also used by De Concini and Procesi 51] and Yuzvinsky 142] for constructing rational models of the cohomology algebra of an arrangement complement. As we see, in the case of coordinate subspace arrangements calculations

MP

MP :


8.2. COORDINATE SUBSPACE ARRANGEMENTS

131

become shorter and more e ective as soon as the Stanley{Reisner ring is brought into the picture. Problem 8.14. Calculate the integer cohomology algebra of a coordinate subspace arrangement complement and compare it with the corresponding Tor-algebra TorZ v1;::: ;vm ] (Z(K ); Z). Example 8.15. Let K be a disjoint union of m vertices. Then U (K ) is the complement to the set of all codimension-two coordinate subspaces zi = zj = 0, 1 6 i < j 6 m, in C m (see Example 8.7). The face ring is k(K ) = k v1 ; : : : ; vm ]=IK , where IK is generated by the monomials vi vj , i 6= j . An easy calculation using Corollary 8.13 shows that the subspace of cocycles in k(K ) u1; : : : ; um] has 6 the basis consisting of monomials vi1 ui2 ui3 uik with k > 2 and ip = iq for p 6= q. Since deg(vi1 ui2 ui3 uik ) = k + 1, the space of (k + 1)-dimensional cocycles ?? ? has dimension m m?11 . The space of (k + 1)-dimensional coboundaries is m k k dimensional (it is spanned by the coboundaries of the form d(ui1 uik )). Hence, ? ? ? dim H 0 U (K ) = 1; H 1 U (K ) = H 2 U (K ) = 0; ?? ? ? ? dim H k+1 U (K ) = m m?11 ? m = (k ? 1) m ; 2 6 k 6 m; k k k and the multiplication in the cohomology is trivial. In particular, for m = 3 we have 6 three-dimensional cohomology classes vi uj ], i 6= j , sub ject to 3 relations vi uj ] = vj ui ], and 3 four-dimensional cohomology classes v1 u2 u3 ], v2 u1 u3 ], v3 u1 u2 ] sub ject to one relation v1 u2u3 ] ? v2 u1 u3 ] + v3 u1 u2 ] = 0: 3 (U (K )) = 3, dim H 4 (U (K )) = 2, and the multiplication is trivial. Hence, dim H It can be shown that U (K ) in this case has a homotopy type of a wedge of spheres: U (K ) ' S 3 _ S 3 _ S 3 _ S 4 _ S 4 : Example 8.16. Let K be the boundary of an m-gon, m > 3. Then U (K ) = C m n fzi = zj = 0g: By Theorem 8.13, the cohomology ring of H (U (K ); k) is isomorphic to the ring described in Example 7.22 (note that the multiplication is non-trivial here). As it is shown in 65], in the case of arrangements of real coordinate subspaces only additive analogue of our Theorem 8.13 holds. Namely, let us consider the polynomial ring k x1 ; : : : ; xm ] with deg xi = 1, i = 1; : : : ; m. Then the graded structure in the face ring k(K ) changes accordingly. The Betti numbers of the real coordinate subspace arrangement UR(K ) can be calculated by means of the following result. Theorem 8.17 ( 65, Theorem 3.1]). The fol lowing isomorphism holds: X ? ? H p UR(K ) = Tor?i;j ;::: ;xm] k(K ); k = H ?i;j u1; : : : ; um] k(K ); d ; k x1
where bideg ui = (?1; 1), bideg vi = (0; 1), dui = xi , dxi = 0. As it was observed in 65], there is no multiplicative isomorphism analogous to Theorem 8.13 in the case of real arrangements, that is, the algebras H (UR(K )) and Tork x1;::: ;xm ] (k(K ); k) are not isomorphic in general. The paper 65] also contains
?i+j=p
i?j 6=0;1 mod m


132

8. COHOMOLOGY RINGS OF SUBSPACE ARRANGEMENT COMPLEMENTS

the formulation of the rst multiplicative isomorphism of our Theorem 8.13 for complex coordinate subspace arrangements (see 65, Theorem 3.6]), with reference to a paper by Babson and Chan (unpublished). Up to this point we have used the description of coordinate subspaces by means of equations (see (8.2)). On the other hand, a coordinate subspace can be de ned as the linear span of a subset of the standard basis fe 1 ; : : : ; e m g. This leads to the dual approach to coordinate subspace arrangements, which corresponds to the b passage from simplicial complex K to the dual complex K (Example 2.26). Namely, we have b C A(K ) = spanfei1 ; : : : ; eik g : fi1; : : : ; ik g 2 K (see Construction 8.5). We may observe further that in the coordinate subspace arrangement case the intersection poset (L; <) is the inclusion poset of simplices b b of K with added maximal element (or equivalently, the inclusion poset of cone K ). Hence, ord(L(v;T ) ) is the barycentric subdivision of linkK v, where v is regarded as b b a simplex of K . Thus, we may rewrite the Goresky{MacPherson formula (8.1) in the complex subspace arrangement case as M ? ? e e (8.7) Hi U (K ) = H 2m?2j j?i?2 linkK ; b (note that d( ) = j j and the dimensions are doubled since we are in the complex arrangement case). The above observations were used in 54] to describe the product of two cohomology classes of a coordinate subspace arrangement complement (either real or b complex) in terms of the combinatorics of links of simplices in K (see 54, Theorem 1.1]). On the other hand, the isomorphism of algebras established in Theorem 8.13 allows us to connect two seemingly unrelated results, namely, the Goresky{MacPherson theorem for the cohomology of an arrangement complement and the Hochster theorem from the commutative algebra. Proposition 8.18. After identi cation of the cohomology H (U (K )) with the Tor-algebra Tork v1 ;:::;vm ] (k(K ); k) established by Theorem 8.13, the Hochster Theorem 3.27 becomes equivalent to the Goresky{MacPherson Theorem 8.4 in the case of coordinate subspace arrangements. Proof. Using Theorem 8.13 to identify ?i;2j (k(K )) with dimk H ?i;2j (U (K )), we get the following formula from Hochster's Theorem 3.27: M ? e Hp?j j?1(K ): Hp U (K ) = Non-empty simplices 2 K do not sponding full subcomplexes K are subset of m] only contributes k to formula as ? e (8.8) Hp U (K ) contribute to the above sum since the corree contractible. Since H?1 (?) = k, the empty H0 (U (K )). Hence, we may rewrite the above =
M b 2K

m]

Using the Alexander duality (Proposition 2.29), we calculate ? ? e e e Hp?j j?1(K ) = H m?3?p+j j+1?jbj linkK b = H 2m?2jbj?p?2 linkK b ; b b

= 2K

e Hp?j j?1(K ):


8.3. DIAGONAL SUBSPACE ARRANGEMENTS

133

b where b = m] n is a simplex of K . Now we observe that (8.8) is equivalent to (8.7).

Another interesting particular class of subspace arrangements is diagonal arrangements. A classical example of a diagonal subspace arrangement is given by the arrangement of all diagonal hyperplanes fzi = zj g in C m , mentioned in section 8.1 (see Example 8.3). Some further particular examples of diagonal arrangements, the so-called k-equal arrangements were considered, e.g. in 20], while the cohomology of general diagonal arrangement complements was studied in 114]. In this section we establish certain relationships between this cohomology and the cohomology of the loop spaces (BT ZK ) (see section 6.5) and ZK . Definition 8.19. For each subset = fi1 ; : : : ; ik g m] de ne the diagonal subspace D in Rm by D = f(y1; : : : ; ym) 2 Rm : yi1 = = yik g: Diagonal subspaces in C m are de ned similarly. An arrangement A = fL1; : : : ; Lr g is called diagonal if all Li , i = 1; : : : ; r, are diagonal subspaces. Construction 8.20. Given a simplicial complex K on the vertex set m], introduce the diagonal subspace arrangement DA(K ) as the set of subspaces D such that is not a simplex of K : DA(K ) = fD : 2 K g: = Denote the complement of the arrangement DA(K ) by M (K ). The following statement is proved in the similar way as the corresponding statement (Proposition 8.6) for coordinate subspace arrangements. Proposition 8.21. The assignment K 7! M (K ) de nes a one-to-one orderpreserving correspondence between the set of simplicial complexes on the vertex set m] and the set of diagonal subspace arrangement complements in Rm . Here we still assume that k is a eld. The multigraded (or N m -graded) structure in the ring k v1 ; : : : ; vm ] (Construction 3.33) de nes an Nm -grading in the i im Stanley{Reisner ring k(K ). The monomial v11 vm acquires the multidegree (2i1; : : : ; 2im). Let us consider the modules Tork(K ) (k; k). They can be calculated by means of the minimal free resolution (Example 3.23) of k (regarded as a k(K )-module). The minimal resolution also carries a natural Nm -grading, and we denote the subgroup of elements of multidegree (2i1 ; : : : ; 2im) in Tork(K ) (k; k) by Tork(K )(k; k)(2i1 ;::: ;2im ) . Theorem 8.22 ( 114, Theorem 1.3]). The fol lowing isomorphism holds for the cohomology groups of a real diagonal subspace arrangement complement M (K ): ? m H i M (K ); k = Tor?((K )?i) (k; k)(2;::: ;2) : k
Remark. Instead of simplicial complexes K on the vertex set m] the authors of 114] considered square-free monomial ideals I k v1 ; : : : ; vm ]. Proposition 3.3 shows that the two approaches are equivalent.

8.3. Diagonal subspace arrangements and the cohomology of ZK .


134

8. COHOMOLOGY RINGS OF SUBSPACE ARRANGEMENT COMPLEMENTS

bration P ! DJ (K ) with bre DJ (K ), where DJ (K ) is the Davis{Januszkiewicz space (De nition 6.27) and P is the path space over DJ (K ). By Corollary 7.4, ? E2 = TorH (DJ (K )) H (P ); k = Tork(K )(k; k); (8.9) and the spectral sequence converges to TorC (DJ (K ))(C (P ); k) = H ( DJ (K )). Since P is contractible, there is a cochain equivalence C (P ) ' k. We have C (DJ (K )) = k(K ). Therefore, ? TorC (DJ (K )) C (P ); k = Tork(K ) (k; k); which together with (8.9) shows that the spectral sequence collapses at the E2 term. Hence, H ( DJ (K )) = Tork(K ) (k; k). Finally, Theorem 6.29 shows that H ( DJ (K )) = H ( BT ZK ), which concludes the proof. Proposition 8.24. The fol lowing isomorphism of algebras holds ? H (BT ZK ) = H ( ZK ) u1 ; : : : ; um ]: Proof. Consider the bundle BT ZK ! B T m with bre ZK . It is easy to see that the corresponding loop bundle BT ZK ! T m with bre ZK is trivial (note that B T m ' T m). To nish the proof it remains to mention that H (T m) = u1; : : : ; um]. Theorems 8.9 and 8.13 give an application of the theory of moment-angle complexes to calculating the cohomology ring of a coordinate subspace arrangement complement. Likewise, Theorems 8.22, 8.23 and Proposition 8.24 establish a connection between the cohomology of a diagonal subspace arrangement complement and the cohomology of the loop space over the moment-angle complex ZK . However, the latter relationships are more subtle than those in the case of coordinate subspace arrangements. For instance, we do not have an analogue of the multiplicative isomorphism from Theorem 8.13. It would be very interesting to get any statement of such kind, or discover other new applications of the theory of momentangle complexes to diagonal (or maybe even general) subspace arrangements.

Theorem 8.23. The fol lowing additive isomorphism holds: ? H (BT ZK ); k = Tork(K ) (k; k): Proof. Let us consider the Eilenberg{Moore spectral sequence of the Serre -


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Index

A ne equivalence, 7 Alexander duality, 26 simplicial, 27 Ample divisor, 61 Arithmetic genus, 80 Arnold relations, 126 Arrangement, 125 diagonal, 133 coordinate, 127 hyperplane, 125 k-equal, 133 subspace (central), 125 Artin group, 97 Bistellar equivalence, 32 moves ( ips, operations), 29, 32, 58, 93 Barycenter, 24 Barycentric subdivision, 24 Bigraded Betti numbers, 42 di erential module, 40 di erential algebra, 41, 42 Blow-up, blow-down, 31, 58 Borel construction, 94 Boundary, 7 Bounded ag manifold, 72 Chain, 8 Characteristic map, 64 directed, 71 Characteristic pair, 65 directed, 71 Charney{Davis conjecture, 80 Chow ring, 59 Chromatic number, 114 Cobordisms
141

complex, 70 oriented, 75 Cohen{Macaulay complex, 39, 49, 106 ring (algebra), 38 Colimit, 97 coloring, 99, 114 Combinatorial equivalence, 7, 23 neighborhood, 26 Complement (of an arrangement), 125 Cone, 23 convex polyhedral, 57 non-singular, 58 rational, 58 simplicial, 58 strongly convex, 58 Connected sum of simple polytopes, 10 of simplicial complexes, 24 Convex polyhedron, 7 Core, 26 Coxeter group, 2, 97 Coxeter complex, 2 Cross-polytope, 10 Cube, 8 standard, 8 topological, 49 Cubical complex, 49 abstract, 49 combinatorial-geometrical, 49 embeddable into lattice, 50 subdivision, 54 of simple polytope, 51


142

8.3. INDEX

Davis{Januszkiewicz space, 95 Dehn{Sommerville relations, 13, 47, 62, 112 for triangulated manifolds, 120 Dehn twist, 99 Depth, 38 Dimension homological, 40 Krull, 38 of cubical complex, 49 of polytope, 7 of simplicial complex, 21 Dolbeault complex, 75 Edge, 7 Edge vector, 76 Elementary shellings, 33 Eulerian complex, 47 Face missing, 25 of convex polyhedral cone, 58 of cubical complex, 49 of polytope, 7 of polyhedron, 22 proper, 7 Face poset, 8 Face ring of simple polytope, 20 of simplicial complex, 35 Facet, 7 Facet vector, 64 Facial submanifold, 64 Factorization conjecture (strong, weak), 58 Fan, 57 complete, 58 non-singular, 58 normal, 60 polytopal, 67 simplicial, 58 strongly polytopal, 67 weakly polytopal, 67 Four Color Theorem, 99, 114 Flag complex, 25 Flagi cation, 25 Flip, 58 f -vector of cubical complex, 49 of polytope, 12 of simplicial complex, 22

ring (algebra), 46 Graph product, 97

h-vector of algebra, 38 of polytope, 12 of simplicial complex, 22 Hard Lefschetz theorem, 61, 113 Hauptvermutung der Topologie, 30 Hilbert series, 37 Hinge mechanisms, 98 Hirzebruch genus, 75 Hirzebruch surface, 83 Hopf conjecture, 80 Homotopy colimit, 97, 127 hsop (homogeneous system of parameters), 38 Ideal monomial, 36 Stanley{Reisner, 35 Index (of a vertex), 77 Intersection cohomology, 61 Intersection h-vector, 62 Intersection poset (of an arrangement), 125
Join, 23

L-genus, 75 Link, 25 lsop (linear system of parameters), 38 Lower Bound Conjecture (LBC), 17 generalized (GLBC), 20, 29, 62, 113
Manifold P L (combinatorial), 29 quasitoric, 63, 64 non-toric, 68 stably complex, 69 toric, 58 unitary, 81 triangulated (simplicial), 29 with corners, 63 Milnor hypersurfaces, 70, 73 Milnor ltration, 99 Minimal generator set (basis), 41 map, 41 resolution, 41 Mirroring construction, 98 Moment-angle manifold, 87 complex 2, 88 Moment curve, 11 Moment map, 63, 130 Morse theory, 13, 66, 113 strati ed, 126 Multi-fan, 81 M -vector, 17

g-conjecture, 29 g-theorem, 16, 47, 61, 113 g-vector of polytope, 12 of simplicial complex, 22 Geometrical realization, 22 Ghost vertex, 88 Gorenstein, Gorenstein* complex, 46, 80, 89, 113


8.3. INDEX

143

Noether normalization lemma, 38 Omniorientation, 70 Orbifold, 58, 98 Order complex, 25 of arrangement, 125 Oriented matroid, 126 Piecewise linear (P L) homeomorphism, 23 manifold, 29 with boundary, 33 map, 23 sphere, 28 Poincare series, 37 Polar set, 9 Polyhedron, 21 convex, 7 Polytope combinatorial, 8 convex, 7 cyclic, 11 generic, 9 geometrical, 8 k-neighborly, 11 lattice, 60 neighborly, 11 non-rational, 60, 62 polar (dual), 10 rational, 60 simple, 9 simplicial, 9 stacked, 19 Polytope algebra, 18 Poset, 8 Eulerian, 47 Poset category, 127 Positive cone, 8 Product of simple polytopes, 10 of simplicial complexes, 24 Pseudomanifold, 111 Pullback from the linear model, 98 Quadratic algebra, 36 Rank function, 125 Regular sequence, 38 Resolution free, 40 Koszul, 41 minimal, 41 pro jective, 42 Ray, 58 Schlegel diagram, 28 Serre problem, 42 Sign (of a vertex), 76 Signature, 75, 79 Simplex, 8

abstract, 21 geometrical, 21 standard, 8 regular, 8 Simplicial complex, 21 abstract, 21 dual, 26 geometrical, 21 k-neighborly, 96 pure, 21 underlying (of a fan), 58 fan, 58 isomorphism, 23 manifold, 29 map, 23 non-degenerate, 23 sphere, 28 subdivision, 23 of cube, 51 stellar, 31 Skeletal rigidity, 29 Small cover, 98 Spectral sequence Anderson, 127 Eilenberg{Moore, 101, 103, 116, 134 Leray{Serre, 74, 96, 104, 115 Sphere Barnette, 28, 68 Bruckner, 28 homology, 28 non-P L, 28, 30 non-polytopal, 28 P L, 28 Poincare, 30 polytopal, 28 simplicial, 28 stacked, 32 Stably complex manifold, 69 structure, 69 canonical, 70 Stanley{Reisner ring of simple polytope, 20 of simplicial complex, 35 Star, 25 Subcomplex full, 26 cubical, 50 simplicial, 21 Support (of an arrangement), 125 Supporting hyperplane, 7 Surgery, 31 equivariant, 93 Suspension, 23 Symplectic reduction, 130 Tangent bundle, 69 T n -manifold, 63


144

8.3. INDEX

Todd genus, 75, 80 Tor-algebra, 44 Toric variety, 31, 57 Torus, 57 algebraic, 57 Torus action Hamiltonian, 130 locally standard, 64 standard, 63 Triangulation Conjecture, 31 Union (of an arrangement), 125 Upper Bound Conjecture (UBC) for polytopes, 18 for simplicial spheres, 39 Vertex, 7, 21 Vertex set, 21 Volume polynomial, 18 2-arrangement, 126 y -genus, 75 -equivariant, 63, 65 -translation, 65