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TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND
HOMOLOGICAL ALGEBRA
VICTOR M. BUCHSTABER AND TARAS E. PANOV
Abstract. The paper surveys some new results and open problems connected
with such fundamental combinatorial concepts as polytopes, simplicial com­
plexes, cubical complexes, and subspace arrangements. Particular attention is
paid to the case of simplicial and cubical subdivisions of manifolds and, espe­
cially, spheres. We describe important constructions which allow to study all
these combinatorial objects by means of methods of commutative and homo­
logical algebra. The proposed approach to combinatorial problems relies on the
theory of moment­angle complexes, currently being developed by the authors.
The theory centres around the construction that assigns to each simplicial com­
plex K with m vertices a T m ­space ZK with a special bigraded cellular decom­
position. In the framework of this theory, the well­known non­singular toric
varieties arise as orbit spaces of maximally free actions of subtori on moment­
angle complexes corresponding to simplicial spheres. We express different in­
variants of simplicial complexes and related combinatorial­geometrical objects
in terms of the bigraded cohomology rings of the corresponding moment­angle
complexes. Finally, we show that the new relationships between combinatorics,
geometry and topology result in solutions to some well­known topological prob­
lems.
Contents
Introduction 2
1. Algebraic, combinatorial and geometrical background 3
1.1. Polytopes 3
1.2. Simplicial complexes: topology and combinatorics 12
1.3. Simplicial complexes: commutative algebra 16
1.4. Homological properties of face rings (Stanley--Reisner rings) 19
1.5. Cubical complexes and cubical maps 24
2. Toric and quasitoric manifolds 29
2.1. Toric varieties 29
2.2. Quasitoric manifolds 33
2.3. Stably complex structures and quasitoric representatives in cobordism
classes 38
2.4. Combinatorial formulae for Hirzebruch genera of quasitoric manifolds 41
2.5. The classification problem for quasitoric manifolds over a given simple
polytope 47
3. Moment­angle complexes 48
3.1. Moment­angle manifolds ZP defined by simple polytopes 48
3.2. General moment­angle complexes 51
1991 Mathematics Subject Classification. 52B70, 57Q15, 57R19, 14M25, 52B05, 13F55, 05B35.
Partially supported by the Russian Foundation for Fundamental Research, grant no. 99­01­
00090.
1

2 VICTOR M. BUCHSTABER AND TARAS E. PANOV
3.3. Cellular structures on moment­angle complexes 52
3.4. Borel construction and Stanley--Reisner space 54
3.5. Generalisations, analogues and additional comments 56
4. Cohomology of moment­angle complexes and combinatorics of simplicial
manifolds 58
4.1. Eilenberg--Moore spectral sequence 58
4.2. Cohomology of moment­angle complex ZK : the case of general K 60
4.3. Cohomology of moment­angle complex ZK : the case of spherical K 66
4.4. Partial quotients of manifold ZP 69
4.5. Bigraded Poincar'e duality and analogues of Dehn--Sommerville
equations for simplicial manifolds 73
5. Subspace arrangements and cohomology rings of their complements 77
5.1. Summary of results on the cohomology of general arrangement
complements 77
5.2. Coordinate subspace arrangements and cohomology of ZK . 79
5.3. Diagonal subspace arrangements and cohomology of loop
space\Omega ZK . 84
References 85
Introduction
We survey the results and open problems in the vast field, which was being shaped
during the last two decades and incorporated various aspects of combinatorics of
polytopes, combinatorial and algebraic topology, homological algebra, group actions
on topological spaces, algebraic geometry of toric varieties and symplectic geometry.
The main aim of this review is to show that the theory of moment­angle complexes
proposed by the authors allows to substantially extend the relationships between
the above listed branches of mathematical science and thereby obtain solutions to
some well­known problems. Each section of the survey refers to a separate subject
and contains the necessary introductory remarks. Below we schematically overview
the contents of the article; this can be considered as the guide to the whole survey.
Chapter 1 contains the necessary combinatorial, geometrical and topological facts
about polytopes, simplicial and cubical complexes and manifolds. We describe both
classical and original constructions, which allow to study the combinatorial objects
by methods of commutative and homological algebra.
Section 1 of chapter 2 is the overview of constructions and results on algebraic
toric varieties and related combinatorial objects, which are necessary for the rest
of the paper. In section 2.2 we describe the topological analogues of toric varieties,
the quasitoric manifolds introduced by Davis and Januszkiewicz. The main con­
struction and results about the quasitoric manifolds can be also found there. In
section 2.3 we present the solution to the quasitoric analogue of the Hirzebruch
problem about connected representatives in the cobordism classes of stably com­
plex manifolds, which was recently obtained by Buchstaber and Ray. In section 2.4
we give the obtained by Panov combinatorial formulae for Hirzebruch genera of
quasitoric manifolds. The last section of chapter 2 describes the known results on
the classification of toric and quasitoric manifolds over a given simple polytope.
The theory of moment­angle complexes being developed by the authors is the
subject of chapters 3 and 4. To each simplicial complex K with m vertices there is

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 3
assigned the moment­angle complex ZK (see section 3.2). The complex ZK caries
a canonical action of the torus T m with quotient the cone over K and possesses the
canonical bigraded cellular decomposition (see section 3.3). A series of important
results of the theory arises from the fact that ZK is a manifold provided that K is a
simplicial sphere. At the same time in the more general case of simplicial manifold
K the singular points of ZK form an orbit of the torus action, and the complement
of the neighbourhood of this orbit is a manifold with boundary. Using the bigraded
cellular structure, in chapter 4 we calculate the bigraded cohomology ring of the
moment­angle complex ZK . This calculation reveals new relationships with some
well­known constructions from homological algebra and opens the way to solution
of many combinatorial problems.
In chapter 5 we apply the theory of moment­angle complexes to the well­known
problem of calculation the cohomology of subspace arrangement complements. We
concentrate on coordinate subspace arrangements and diagonal subspace arrange­
ments (sections 5.2 and 5.3 correspondingly) and survey the results obtained by
the authors in this direction. In particular, we calculate the cohomology ring of the
complement of a coordinate subspace arrangement by reducing to the cohomology
of the moment­angle complex ZK . We also reduce the problem of calculating the
cohomology of the diagonal subspace arrangement complement to calculating the
cohomology of the loop
space\Omega ZK .
Almost all new concepts in our survey are provided by the corresponding exam­
ples. We also give a lot of examples of particular computations, which illustrate the
general theorems. All the results in our paper are of three types. Firstly, interpre­
tations of classical results, which sometimes are given without references. Secondly,
results of other authors, which are always provided by the corresponding references.
And thirdly, results that either have been obtained recently by the present authors,
or are extensions of results from the authors papers [19]--[23], [68], [69] and papers
[25], [26] by N. Ray and the first author.
The authors wish to express special thanks to Levan Alania, Yusuf Civan, Na­
talia Dobrinskaya, Nikolai Dolbilin, Mikhail Farber, Ivan Izmestiev, Oleg Musin,
Sergey Novikov, Nigel Ray, Elmer Rees, Mikhail Shtan'ko, Mikhail Shtogrin,
Vladimir Smirnov, Neil Strickland, Sergey Tarasov, Victor Vassiliev, Volkmar Wel­
ker, Sergey Yuzvinsky, G¨unter Ziegler for the insight gained from numerous discus­
sions of different questions relevant to the survey. We also grateful to all participants
of the seminar ``Topology and computational geometry'' being held by O. R. Musin
and the authors at the Department of Mathematics and Mechanics, Moscow State
University.
1. Algebraic, combinatorial and geometrical background
1.1. Polytopes. Both combinatorial and geometrical aspects of the theory of con­
vex polytopes are exposed in a vast number of textbooks, monographs and papers.
We just mention Ziegler's book [89], where a host of further references can be found.
In this section we review some basic concepts and constructions used in the rest of
the paper.
There are two different ways to define a convex polytope in n­dimensional affine
space R n .
Definition 1.1.1. A convex polytope is the convex hull of a finite set of points in
some R n .

4 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Definition 1.1.2. A convex polyhedron P is an intersection of finitely many half­
spaces in some R n :
P =
\Phi
x 2 R n : hl i ; x i – \Gammaa i ; i = 1; : : : ; m
\Psi
;
(1)
where l i 2 (R n ) \Lambda , i = 1; : : : ; m, are some linear functions and a i 2 R. A (convex)
polytope is a bounded convex polyhedron.
Nevertheless, the above two definitions produce the same geometrical object, i.e.
the subset of R n is a convex hull of a finite point set if and only if it is a bounded
intersection of finitely many half­spaces. This fact is proved in many textbooks on
polytopes and convex geometry, see e.g. [89, Theorem 1.1].
The dimension of a polytope is the dimension of its affine hull; without loss
of generality we may consider only n­dimensional polytopes P n in n­dimensional
space R n . A supporting hyperplane of P n is an affine hyperplane H which intersects
P n and for which the polytope is contained in one of the two closed half­spaces
determined by the hyperplane. The intersection P n `` H is 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 n­dimensional
polytope P n is itself a polytope of dimension Ÿ n. 0­dimensional faces are called
vertices, 1­dimensional faces are called edges, and codimension one faces are called
facets. The faces of P n of all dimensions form a partially ordered set (poset) with
respect to the inclusion. This poset is called the face lattice of P n .
Two polytopes P 1 2 R n1 and P 2 2 R n2 of the same dimension are said to
be affinely equivalent if there is an affine map R n1 ! R n2 taking one polytope to
another. Two polytopes are combinatorially equivalent if there is a bijection between
their faces that preserves the inclusion relation. In other words, two polytopes are
combinatorially equivalent if their face lattices are isomorphic as posets.
Example 1.1.3 (simplex and cube). An n­dimensional simplex \Delta n is the convex
hull of (n + 1) points of R n that do not lie on a common affine hyperplane.
All faces of an n­simplex are simplices of dimension Ÿ n. All n­simplices are
affinely equivalent. The standard n­simplex is the convex hull of points (1; 0; : : : ; 0),
(0; 1; : : : ; 0); : : : ; (0; : : : ; 0; 1), and (0; : : : ; 0) in R n . Alternatively, the standard n­
simplex is defined by (n + 1) inequalities
x i – 0; i = 1; : : : ; n; and \Gamma x 1 \Gamma : : : \Gamma xn – \Gamma1:
(2)
The regular n­simplex is the convex hull of n + 1 points (1; 0; : : : ; 0), (0; 1; : : : ; 0),
: : : , (0; : : : ; 0; 1) in R n+1 .
The standard q­cube is the convex polytope I q ae R q defined by
I q = f(y 1 ; : : : ; y q ) 2 R q : 0 Ÿ y i Ÿ 1; i = 1; : : : ; qg:
(3)
Alternatively, the standard q­cube is the convex hull of 2 q points in R q having only
zero and unit coordinates.
The following construction identifies a convex n­polytope with m facets as the
intersection of the positive cone
R m
+ =
\Phi
(y 1 ; : : : ; y m ) 2 R m : y i – 0; i = 1; : : : ; m
\Psi ae R m
(4)
with a certain n­dimensional plane.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 5
Construction 1.1.4. Let P 2 R n be a convex n­polytope given by (1) with some
l i 2 (R n ) \Lambda , a i 2 R, i = 1; : : : ; m. Introduce n \Theta m­matrix L whose columns are
vectors l i written in the standard basis of (R n ) \Lambda , i.e. (L) ji = (l i ) j . Note that L is of
rank n. Likewise, let a = (a 1 ; : : : ; am ) t 2 R m be the column vector with entries a i .
Then (1) shows that
P =
\Phi
x 2 R n : (L t x + a) i – 0; i = 1; : : : ; m
\Psi ;
(5)
where L t is the transposed matrix and x = (x 1 ; : : : ; xn ) t is the column vector.
Consider the affine map
AP : R n ! R m ; AP (x ) = L t x + a 2 R m :
(6)
Its image is an n­dimensional plane in R m , and (5) shows that AP (P ) is the intersec­
tion of this plane with the positive cone R m
+ . Now, let W be an m \Theta (m \Gamma n)­matrix
of rank (m \Gamma n) such that LW = 0. Then it is easy to see that
AP (P ) =
\Phi
y 2 R m : W t y = W t a; y i – 0; i = 1; : : : ; m
\Psi :
Note that the polytopes P and AP (P ) are affinely equivalent.
Example 1.1.5. Consider the standard n­simplex \Delta n ae R n defined by inequali­
ties (2). It has m = n + 1 facets, and l 1 = (1; 0; : : : ; 0) t , : : : , l n = (0; : : : ; 0; 1) t ,
l n+1 = (\Gamma1; : : : ; \Gamma1) t , a 1 = : : : = an = 0, an+1 = 1 (see (1)). One can take
W = (1; : : : ; 1) t in Construction 1.1.4. Hence, W t y = y 1 + : : : + ym , W t a = 1, and
we have
A \Delta n (\Delta n ) =
\Phi
y 2 R n+1 : y 1 + : : : + y n+1 = 1; y i – 0; i = 1; : : : ; n
\Psi
:
This is the regular n­simplex in R n+1 .
Remark. Convex polytopes introduced in definitions 1.1.1 and 1.1.2 are geometrical
objects. However, one could also consider combinatorial polytopes which are classes
of combinatorially equivalent polytopes. In fact, a combinatorial polytope is the face
lattice (regarded as a poset) of a (geometrical) polytope. In this review we deal with
both geometrical and combinatorial polytopes.
Two different definitions of a convex polytope lead to two different notions of
generic polytopes.
A set of m ? n points in R n is in general position if no (n + 1) of them lie on
a common affine hyperplane. From the viewpoint of Definition 1.1.1, the convex
polytope is generic if it is the convex hull of a set of points in general position. 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 .
A set of m ? n hyperplanes hl i ; xi = \Gammaa i , l i 2 (R n ) \Lambda , x 2 R n , a i 2 R, i =
1; : : : ; m, is in general position if no point lies in more than n of them. From the
viewpoint of Definition 1.1.2, the convex polytope P n is generic if its bounding
hyperplanes (see (1)) are in general position, i.e. there exactly n facets meet at
each vertex of P n . Such polytopes are called simple.
For any convex polytope P ae R n define its polar set P \Lambda ae (R n ) \Lambda as
P \Lambda = fx 0 2 (R n ) \Lambda : hx 0 ; x i – \Gamma1 for all x 2 Pg:
Remark. Our definition of the polar set is that used in the algebraic geometry of
toric varieties, not the classical one from the convex geometry. The latter is obtained
by replacing the inequality ``– \Gamma1'' above by ``Ÿ 1''. One polar set is taken into
another by the symmetry with centre in 0.

6 VICTOR M. BUCHSTABER AND TARAS E. PANOV
It is well known in convex geometry that the polar set P \Lambda is a convex polytope
in the dual space (R n ) \Lambda and (P \Lambda ) \Lambda = P provided that 0 2 P . The polytope P \Lambda
is called the polar (or dual) of P . Moreover, there is a one­to­one order reversing
correspondence between face lattices of P and P \Lambda . In particular, if P is simple, then
P \Lambda is simplicial, and vice versa.
Example 1.1.6. Any polygon (2­polytope) is simple and simplicial at the same
time. In dimensions – 3 only polytope that is simultaneously simple and simplicial
is the simplex. The cube is a simple polytope. The polar of a simplex is again
a simplex. The polar of the cube is called the cross­polytope. The 3­dimensional
cross­polytope is the octahedron.
In the sequel simple polytopes are denoted by the Latin letters P , Q etc., while
simplicial ones are denoted by the same letters with asterisque: P \Lambda , Q \Lambda etc.
Each face of a simple polytope is a simple polytope. The product P 1 \Theta P 2 of
two simple polytopes P 1 and P 2 is a simple polytope as well. Let P \Lambda
1 and P \Lambda
2
be the corresponding polar simplicial polytopes. Then P \Lambda
1 ffi P \Lambda
2 := (P 1 \Theta P 2 ) \Lambda is
again a simplicial polytope. The operation ffi on simplicial polytopes can be directly
described as follows. Realise P \Lambda
1 in R n1 and P \Lambda
2 in R n2 in such way that 0 2 P \Lambda
1 and
0 2 P \Lambda
2 . Then P \Lambda
1 ffi P \Lambda
2 ae R n1 \Theta R n2 is the convex hull of the union of P \Lambda
1 ae R n1 \Theta 0
and P \Lambda
2 ae 0 \Theta R n2 .
Construction 1.1.7 (Connected sum of two simple polytopes). Suppose we are
given two simple polytopes P n and Q n , both of dimension n, with distinguished
vertices v and w respectively. The informal way to get the connected sum P n # v;w Q n
of P n at v and Q n at w is as follows. We ``cut off'' v from P n and w from Q n ; then,
after a projective transformation, we can ``glue'' the rest of P n to the rest of Q n
along the new facets to obtain P n # v;w Q n . Below we give the formal definition,
following [26, x6]; this definition will be used later.
First, we introduce an n­polyhedron \Gamma n , which will be used as a template for
the construction; it arises by considering the standard (n \Gamma 1)­simplex \Delta n\Gamma1 in
the subspace fx : x 1 = 0g of R n , and taking its cartesian product with the first
coordinate axis. The facets G r of \Gamma n therefore have the form R \Theta D r , where D r ,
1 Ÿ r Ÿ n, are the faces of \Delta n\Gamma1 . Both \Gamma n and the G r are divided into positive
and negative halves, determined by the sign of the coordinate x 1 .
We order the facets of P n meeting in v as E 1 ; : : : ; En , and the facets of Q n
meeting in w as F 1 ; : : : ; Fn . Denote the complementary sets of facets by C v and Cw ;
those in C v avoid v, and those in Cw avoid w.
We now choose projective transformations OE P and OE Q of R n , whose purpose is
to map v and w to x 1 = \Sigma1 respectively. We insist that OE P embeds P n in \Gamma n
so as to satisfy two conditions; firstly, that the hyperplane defining E r is identified
with the hyperplane defining G r , for each 1 Ÿ r Ÿ n, and secondly, that the
images of the hyperplanes defining C v meet \Gamma n in its negative half. Similarly, OE Q
identifies the hyperplane defining F r with that defining G r , for each 1 Ÿ r Ÿ
n, but the images of the hyperplanes defining Cw meet \Gamma n in its positive half.
We define the connected sum P n # v;w Q n of P n at v and Q n at w to be the
simple convex n­polytope determined by the images of the hyperplanes defining
C v and Cw and hyperplanes defining G r , r = 1; : : : ; n. It is defined only up to
combinatorial equivalence; moreover, different choices for either of v and w, or either
of the orderings for E r and F r , are likely to affect the combinatorial type. When

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 7
the choices are clear, or their effect on the result irrelevant, we use the abbreviation
P n #Q n .
The related construction of connected sum P #Q \Lambda of a simple polytope P and
a simplicial polytope Q \Lambda is described in [89, Example 8.41].
A simplicial polytope P \Lambda is called k­neighbourly if any k vertices span a face
of P \Lambda . Likewise, a simple polytope P is called k­neighbourly if any k facets of P
have non­empty intersection (i.e. share a common codimension­k face). Obviously,
every simplicial (or simple) polytope is 1­neighbourly. It can be shown ([18, Corol­
lary 14.5], see also Example 1.1.15 below) that if P \Lambda is a k­neighbourly simplicial
n­polytope and k ?
\Theta n
2
\Lambda , then P \Lambda is an n­simplex. In particular, any 2­neighbourly
simplicial 3­polytope is a simplex. However, there exist simplicial n­polytopes with
any number of vertices which are
\Theta n
2
\Lambda
­neighbourly. Such polytopes are called neigh­
bourly . In particular, there exist a simplicial 4­polytope (different from 4­simplex)
any two vertexes of which are connected by an edge.
Example 1.1.8 (neighbourly 4­polytope). Let P = \Delta 2 \Theta \Delta 2 be the product of two
triangles. Then P is a simple polytope, and it is easy to see that any two facets of
P share a common 2­face. Hence, P is 2­neighbourly. The polar P \Lambda is a neighbourly
simplicial 4­polytope.
More generally, it is easy to see that if a simple polytope P 1 is k 1 ­neighbourly and
a simple polytope P 2 is k 2 ­neighbourly, then the product P 1 \Theta P 2 is a min(k 1 ; k 2 )­
neighbourly simple polytope. It follows that (\Delta n \Theta \Delta n ) \Lambda and (\Delta n \Theta \Delta n+1 ) \Lambda provide
examples of neighbourly simplicial 2n­ and (2n+ 1)­polytopes. Another example of
neighbourly polytopes, with arbitrary number of vertices, is as follows.
Example 1.1.9 (cyclic polytopes). Define the moment curve in R n as
x : R \Gamma! R n ; t 7! x (t) = (t; t 2 ; : : : ; t n ) 2 R n :
For any m ? n define the cyclic polytope C n (t 1 ; : : : ; t m ) as the convex hull of m
distinct points x (t i ), t 1 ! t 2 ! : : : ! t m , on the moment curve. It follows from the
Vandermonde determinant identity that no n+1 points on the moment curve lie on a
common affine hyperplane. Hence, C n (t 1 ; : : : ; t m ) is a simplicial n­polytope. It can
be shown (see [89, Theorem 0.7]) that C n (t 1 ; : : : ; t m ) has exactly m vertices x (t i ),
the combinatorial type of cyclic polytope does not depend on the specific choice
of the parameters t 1 ; : : : ; t m , and C n (t 1 ; : : : ; t m ) is a neighbourly simplicial n­
polytope. We denote the combinatorial cyclic n­polytope with m vertices by C n (m).
Let P be a simple n­polytope. Denote by f i the number of faces of P of codi­
mension (i + 1) (i.e. of dimension (n \Gamma i \Gamma 1)). In particular, f 0 is the number of
facets of P , which we will denote m(P ) or just m.
Definition 1.1.10. The integer vector f (P ) = (f 0 ; : : : ; fn\Gamma1 ) is called the f­vector
of simple polytope P . The integer vector (h 0 ; h 1 ; : : : ; hn ) whose components h i are
defined from the equation
h 0 t n + : : : + hn\Gamma1 t + hn = (t \Gamma 1) n + f 0 (t \Gamma 1) n\Gamma1 + : : : + f n\Gamma1
(7)
is called the h­vector of P . Finally, the integer vector (g 0 ; g 1 ; : : : ; g \Theta n
2
\Lambda ), where
g 0 = 1, g i = h i \Gamma h i\Gamma1 , i ? 0, is called the g­vector of P .

8 VICTOR M. BUCHSTABER AND TARAS E. PANOV
We also put f \Gamma1 = 1, which means that the polytope itself is a face of codi­
mension 0. The f­vector and the h­vector determine each other by means of linear
relations, namely
h k =
k
X
i=0
(\Gamma1) k\Gammai
\Gamma n\Gammai
n\Gammak
\Delta
f i\Gamma1 ; f n\Gamma1\Gammak =
n
X
q=k
\Gamma q
k
\Delta
hn\Gammaq ; k = 0; : : : ; n:
(8)
In particular, h 0 = 1 and hn = (\Gamma1) n
\Gamma 1 \Gamma f 0 + f 1 + : : : + (\Gamma1) n fn\Gamma1
\Delta . By Euler's
theorem,
f 0 \Gamma f 1 + \Delta \Delta \Delta + (\Gamma1) n\Gamma1 fn\Gamma1 = 1 + (\Gamma1) n\Gamma1 ;
(9)
which is equivalent to hn = 1 = h 0 . In the case of simple polytopes Euler's theorem
admits the following generalisation.
Theorem 1.1.11 (Dehn--Sommerville relations). The h­vector of any simple n­
polytope is symmetric, i.e.
h i = hn\Gammai ; i = 0; 1; : : : ; n:
There are a lot of different ways to prove the Dehn--Sommerville equation. We
present a proof which uses a Morse­theoretical argument, firstly appeared in [18].
We will return to this argument in chapter 2.
Proof of Theorem 1.1.11. Let P n ae R n be a simple polytope. Choose a linear func­
tion ' : R n ! R which is generic, that is, it takes different values at all vertices
of P n . For this ' there is a vector šš in R n such that '(x ) = hšš; x i. Note that šš is
parallel to no edge of P n . Now we view ' as a height function on P n . Using ', we
make the 1­skeleton 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). For each vertex v of
P n define its index, ind(v), as the number of incident edges that point towards v.
Denote the number of vertices of index i by I š (i). We claim that I š (i) = hn\Gammai .
Indeed, each face of P n has a unique top vertex (a maximum of the height function
' restricted to the face) and a unique bottom vertex (the minimum of '). Let F k
be a k­face of P n , and v F its top vertex. Since P n is simple, there exactly k edges
of F k meet at vF , whence ind(v F ) – k. On the other hand, each vertex of index
q – k is the top vertex for exactly
\Gamma q
k
\Delta faces of dimension k. It follows that f n\Gamma1\Gammak
(the number of k­faces) can be calculated as
fn\Gamma1\Gammak =
X
q–k
\Gamma q
k
\Delta I š (q):
Now, the second identity from (8) shows that I š (q) = hn\Gammaq , as claimed. In
particular, the number I š (q) does not depend on šš. On the other hand, since
ind š (v) = n \Gamma ind \Gammaš (v) for any vertex v, one has
hn\Gammaq = I š (q) = I \Gammaš (n \Gamma q) = h q :
Using (8), we can rewrite the Dehn--Sommerville equations in terms of the f­vector
as follows
f k\Gamma1 =
n
X
j=k
(\Gamma1) n\Gammaj
\Gamma j
k
\Delta
f j \Gamma1 ; k = 0; 1; : : : ; n:
(10)

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 9
The Dehn--Sommerville equations were established by Dehn for n Ÿ 5 in 1905, and
by Sommerville in the general case in 1927 (see [77]) in the form similar to (10).
Example 1.1.12. Let P n1
1 and P n2
2 be simple polytopes. Any face of P 1 \Theta P 2 is
the product of a face of P 1 and a face of P 2 , whence
f k (P 1 \Theta P 2 ) =
n1 \Gamma1
X
i=\Gamma1
f i (P 1 )f k\Gammai\Gamma1 (P 2 ); k = \Gamma1; 0; : : : ; n 1 + n 2 \Gamma 1:
Set h(P ; t) = h 0 +h 1 t + \Delta \Delta \Delta +hn t n . Then it follows from the above formula and (7)
that
h(P 1 \Theta P 2 ; t) = h(P 1 ; t)h(P 2 ; t):
(11)
Example 1.1.13. Let us express the f­vector and the h­vector of the connected
sum P n # Q n in terms of that of P n and Q n . It follows from Construction 1.1.7
that
f i (P n #Q n ) = f i (P n ) + f i (Q n ) \Gamma \Gamma n
i+1
\Delta ; i = 0; 1; : : : ; n \Gamma 2;
fn\Gamma1 (P n #Q n ) = f n\Gamma1 (P n ) + fn\Gamma1 (Q n ) \Gamma 2
(note that
\Gamma n
i+1
\Delta = f i (\Delta n\Gamma1 )). Then it follows from (8) that
h 0 (P n #Q n ) = hn (P n #Q n ) = 1;
h i (P n #Q n ) = h i (P n ) + h i (Q n ); i = 1; 2; : : : ; n \Gamma 1:
Thus, h i , i = 1; : : : ; n \Gamma 1, define integer­valued functions on the set of simple
polytopes which are linear with respect to the connected sum operation.
Problem 1.1.14. Describe all integer­valued functions on the set of simple poly­
topes which are linear with respect to the connected sum operation.
The f­vector of a simplicial polytope P \Lambda is defined as f (P \Lambda ) = (f 0 ; f 1 ; : : : ; fn\Gamma1 ),
where f i is the number of i­faces (i­simplices) of P \Lambda . The h­vector h(P \Lambda ) =
(h 0 ; h 1 ; : : : ; hn ) is determined by identity (7). Note that if P \Lambda is the simplicial
polytope polar to a simple polytope P , then f i (P \Lambda ) = f i (P ). In particular, the
Dehn--Sommerville equations hold for simplicial polytopes as well.
Example 1.1.15. Suppose P \Lambda is a q­neighbourly simplicial n­polytope. Then
f k\Gamma1 (P \Lambda ) =
\Gamma m
k
\Delta , k Ÿ q. From (8) we obtain
h k (P \Lambda ) =
k
X
i=0
(\Gamma1) k\Gammai
\Gamma n\Gammai
k\Gammai
\Delta\Gamma m
i
\Delta =
\Gamma m\Gamman+k\Gamma1
k
\Delta ; k Ÿ q:
(12)
(The last identity is obtained by calculating the coefficient of t k from two sides of
1
(1+t) n\Gammak+1 (1 + t) m = (1 + t) m\Gamman+k\Gamma1 .) Suppose that P \Lambda is not a simplex. Then
m ? n + 1, which together with (12) gives h 0 ! h 1 ! \Delta \Delta \Delta ! h q . It follows from the
Dehn--Sommerville equations that q Ÿ
\Theta n
2
\Lambda .
A natural question arises: which integer vectors may appear as the f­vectors
of simple (or, equivalently, simplicial) polytopes? The Dehn--Sommerville relations
provide a necessary condition.
Proposition 1.1.16 ([52]). The Dehn--Sommerville relations are the most general
linear equations satisfied by the f­vectors of all simple (or simplicial) polytopes.

10 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Proof. In [52] the statement was proved directly, using f­vectors. Instead, we use h­
vectors, which somewhat simplifies the proof. It is sufficient to prove that the affine
hull of the h­vectors (h 0 ; h 1 ; : : : ; hn ) of simple n­polytopes is an
\Theta n
2
\Lambda ­dimensional
plane. Set Q k := \Delta k \Theta \Delta n\Gammak , k = 0; 1 : : : ;
\Theta n
2
\Lambda . Since h (\Delta k ) = 1 + t + \Delta \Delta \Delta + t k , the
formula (11) gives
h(Q k ) = 1 \Gamma t k+1
1 \Gamma t
\Delta 1 \Gamma t n\Gammak+1
1 \Gamma t :
It follows that h(Q k+1 ) \Gamma h(Q k ) = t k+1 + \Delta \Delta \Delta + t n\Gammak\Gamma1 , k = 0; 1; : : : ;
\Theta n
2
\Lambda \Gamma 1.
Therefore, the vectors h(Q k ), k = 0; 1 : : : ;
\Theta n
2
\Lambda , are affinely independent.
We mention also that the identity (9) is the only linear relation satisfied by the
vectors of face numbers of general convex polytopes.
The conditions characterising the f­vectors of simple (or simplicial) polytopes,
now know as the g­theorem, were conjectured by McMullen [58] in 1970 and proved
by Stanley [79] (necessity) and Billera, Lee [13] (sufficiency) in 1980. Besides the
Dehn--Sommerville equations, the g­theorem contains two groups of inequalities, one
linear and one non­linear. To state the g­theorem completely, we need the following
construction. For any two positive integers a, i there exists a unique binomial i­
expansion of a of the form
a =
\Gamma a i
i
\Delta +
\Gamma a i\Gamma1
i\Gamma1
\Delta + \Delta \Delta \Delta +
\Gamma a j
j
\Delta ;
where a i ? a i\Gamma1 ? \Delta \Delta \Delta ? a j – j – 1. Define
a hii =
\Gamma a i +1
i+1
\Delta
+
\Gamma a i\Gamma1 +1
i
\Delta
+ \Delta \Delta \Delta +
\Gamma a j +1
j+1
\Delta
; 0 hii = 0:
Theorem 1.1.17 (g­theorem). An integer vector (f 0 ; f 1 ; : : : ; f n\Gamma1 ) is the f­vector
of a simple n­polytope if and only if the corresponding sequence (h 0 ; : : : ; hn ) deter­
mined by (7) satisfies the following three conditions:
(a) h i = hn\Gammai , i = 0; : : : ; n (the Dehn--Sommerville equations);
(b) h 0 Ÿ h 1 Ÿ \Delta \Delta \Delta Ÿ h \Theta n
2
\Lambda , i.e. g i – 0, i = 0; 1; : : : ;
\Theta n
2
\Lambda .
(c) h 0 = 1, h i+1 \Gamma h i Ÿ (h i \Gamma h i\Gamma1 ) hii , i.e. g i+1 Ÿ g hii
i , i = 1; : : : ;
\Theta n
2
\Lambda \Gamma 1.
Remark. Obviously, the same conditions characterise the f­vectors of simplicial
polytopes.
An integral sequence (k 0 ; k 1 ; : : : ; k r ) satisfying k 0 = 1 and 0 Ÿ k i+1 Ÿ k hii
i for
i = 1; : : : ; r \Gamma 1 is called an M­vector (after M. Macaulay). Conditions (b) and (c)
from the g­theorem imply that the g­vector (g 0 ; g 1 ; : : : ; g \Theta n
2
\Lambda ) of a simple n­polytope
is an M­vector. On the other hand, the notion of M­vector appears in the following
classification result of commutative algebra.
Theorem 1.1.18 (Macaulay). An integral sequence (k 0 ; k 1 ; : : : ; k r ) is an M ­
vector if and only if there exists a commutative graded algebra A = A 0 \PhiA 2 \Phi\Delta \Delta \Delta\PhiA 2r
over a field k = A 0 , generated (as an algebra) by degree­two elements, such that the
dimension of 2i­th graded component of A equals k i , i.e. dim k A 2i = k i , i = 1; : : : ; r.
The proof can be found in [78].
To prove the sufficiency of the g­theorem, Billera and Lee presented a remarkable
combinatorial­geometrical construction of a simplicial polytope with any prescribed
M­sequence as its g­vector. Stanley's proof of the necessity of g­theorem (i.e. that
the g­vector of a simple polytope is an M­vector) relies upon deep results from

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 11
algebraic geometry: the Hard Lefschetz theorem for the cohomology of toric vari­
eties. We give the ideas of Stanley's proof in section 2.1. After 1995 several more
elementary combinatorial proofs of the g­theorem appeared. The first elementary
proof by McMullen [59], which uses the polytope algebra instead of the cohomology
algebra of toric variety, was still very involved. Recently Timorin [85] found much
more simple elementary proof of the g­theorem, which relies on the interpretation of
McMullen's polytope algebra as the algebra of differential operators (with constant
coefficients) vanishing on the volume polynomial of the polytope.
It follows from the results of [13] that the part (b) of the g­theorem, i.e. the
inequalities
h 0 Ÿ h 1 Ÿ \Delta \Delta \Delta Ÿ h \Theta n
2
\Lambda ;
(13)
give the most general linear inequalities satisfied by the f­vectors of simple (or
simplicial) polytopes. These inequalities are now known as the Generalised Lower
Bound Theorem (GLBT) for simple (simplicial) polytopes. During the last two
decades a lot of work was done and progress achieved in extending the Dehn--
Sommerville equations, GLBT, and g­theorem to more general objects than simpli­
cial polytopes. However, there are still a lot of intriguing open problems here. Some
of them are presented in the survey article by Stanley [82]. In the present paper we
also review some related questions (see the comments in the next section).
The g­theorem has the following important corollary.
Theorem 1.1.19 (Upper Bound Theorem (UBT) for simplicial polytopes, [89, Theorem 8.23, Corollary 8.38]).
From all simplicial n­polytopes P \Lambda with m vertices the cyclic polytope C n (m)
(Example 1.1.9) has the maximal number of i­faces, 2 Ÿ i Ÿ n \Gamma 1. That is, if
f 0 (P \Lambda ) = m, then
f i (P \Lambda ) Ÿ f i
\Gamma C n (m)
\Delta ; 2 Ÿ i Ÿ n \Gamma 1:
Note that f i (C n (m)) =
\Gamma m
i+1
\Delta
for 0 Ÿ i !
\Theta n
2
\Lambda
. The above theorem was conjectured
by Motzkin in 1957. It was proved by McMullen in 1970 and motivated him to
conjecture the g­theorem. McMullen showed also that the Upper Bound theorem is
equivalent to the following inequalities for the h­vector h(P \Lambda ) = (h 0 ; h 1 ; : : : ; hn ):
h i (P \Lambda ) Ÿ
\Gamma m\Gamman+i\Gamma1
i
\Delta ; 0 Ÿ i Ÿ
\Theta n
2
\Lambda
(compare this with Example 1.1.15).
To conclude these remarks about polytopes, we introduce an important alge­
braic invariant of a (combinatorial) simple polytope, which appears many times
throughout this review. Let P be a simple n­polytope with m facets F 1 ; : : : ; Fm .
Fix a commutative ring k with unit. Let k[v 1 ; : : : ; v m ] be the polynomial algebra
over k on m generators. We make it a graded algebra by setting deg(v i ) = 2.
Definition 1.1.20. The face ring (or the Stanley--Reisner ring) of a simple poly­
tope P is the quotient ring
k(P ) = k[v 1 ; : : : ; vm ]=I P ;
where IP is the ideal generated by all square­free monomials v i 1 v i 2
\Delta \Delta \Delta v i s , i 1 !
\Delta \Delta \Delta ! i s , such that F i 1
`` \Delta \Delta \Delta `` F i s = ; in P .
Obviously, k(P ) is a graded k­algebra.

12 VICTOR M. BUCHSTABER AND TARAS E. PANOV
We mention that the Stanley--Reisner ring, the f­vector, and the h­vector are
invariants of a combinatorial simple polytope: they depend only on the face lattice
and do not depend on a particular geometrical realisation.
1.2. Simplicial complexes: topology and combinatorics. Let [m] denote the
index set f1; : : : ; mg. For any subset I ae [m] denote by #I its cardinality.
Definition 1.2.1. An (abstract) simplicial complex on the set [m] is a collection
K = fIg of subsets of [m] such that for each I = fi 1 ; : : : ; i k g 2 K all subsets of I
(including ;) also belong to K. Subsets I 2 K are called (abstract) simplices of K.
Similarly one defines a simplicial complex on any set S. One­element subsets from
K are called vertices of K. If K contains all one­element subsets of [m], then we say
that K is a simplicial complex on the vertex set [m]. The dimension of an abstract
simplex I = fi 1 ; : : : ; i k g 2 K is its cardinality minus one, i.e. dimI = #I \Gamma 1.
The dimension of an abstract simplicial complex is the maximal dimension of its
simplices.
Definition 1.2.2. A geometrical simplicial complex (or polyhedron) is a subset
P ae R n represented as a union of simplices of any dimensions in such a way that
the intersection of any two simplices is a face of each. (By simplices here we mean
convex polytopes defined in Example 1.1.3.)
In the sequel we denote by \Delta m\Gamma1 both the abstract simplex (i.e. the simplicial
complex consisting of all subsets of [m]) and the corresponding polyhedron.
Remark. The notion of polyhedron from Definition 1.1.2 is not the same as that
from the above definition. The first 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 meanings became classical in the appropriate science, we do not change their
names. In this review, by ``polyhedron'' we will usually mean a geometrical simplicial
complex. The ``unbounded polytope'' will be referred to as ``convex polyhedron''.
Anyway, it will be always clear from the context which ``polyhedron'' is under
consideration.
Only finite geometrical simplicial complexes (polyhedrons) are considered in this
review. The dimension of a polyhedron is the maximal dimension of its simplices. It
is a classical fact [71] that any n­dimensional abstract simplicial complex K admits
a geometrical realisation jKj as an n­dimensional polyhedron in R 2n+1 (abstract
simplices of K correspond to (polytopal) simplices of jKj). A geometrical realisa­
tion jKj of an abstract simplicial complex K is unique up to a piecewise­linear
homeomorphism.
Construction 1.2.3. One can construct a geometric realisation of a simplicial
complex K on the vertex set [m] in m­dimensional space as follows. Let e i denote
the i­th unit coordinate vector in R m . For each subset I ae [m] denote by \Delta I the
convex hull of vectors e i with i 2 I. Obviously, \Delta I is a (geometric) simplex. Then
one has
jKj =
[
I2K
\Delta I ae R m :
A simplicial map f : jK 1 j ! jK 2 j of two polyhedrons is any mapping of the set
of vertices of jK 1 j to the vertices of jK 2 j, extended linearly on simplices of jK 1 j to

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 13
the whole of jK 1 j. A polyhedron jK 0 j is called a subdivision of polyhedron jKj if
each simplex of jKj is a union of finitely many simplices of jK 0 j. A piecewise linear
(PL) map f : jK 1 j ! jK 2 j is a map that is simplicial between some subdivisions of
jK 1 j and jK 2 j. The standard reference for the PL topology is [74].
Example 1.2.4 (dual simplicial complex). Let K be a simplicial complex on the
set [m]. Suppose that K is not the (m \Gamma 1)­simplex. Define
b
K :=
\Phi I ae [m] : [m] n I =
2 K
\Psi :
Obviously, b
K is a simplicial complex on [m]. It is called the dual of K.
Construction 1.2.5 (join of simplicial complexes). Let K 1 , K 2 be simplicial com­
plexes on [m 1 ] and [m 2 ] respectively. Identify [m 1 ] [ [m 2 ] with [m 1 +m 2 ]. The join
of K 1 and K 2 is the simplicial complex
K 1 \Lambda K 2 :=
\Phi I ae [m 1 +m 2 ] : I = I 1 [ I 2 ; I 1 2 K 1 ; I 2 2 K 2
\Psi
on the set [m 1 +m 2 ].
Example 1.2.6. 1. If K 1 = \Delta m1 \Gamma1 , K 2 = \Delta m2 \Gamma1 , then K 1 \Lambda K 2 = \Delta m1+m2 \Gamma1 .
2. The simplicial complex \Delta 0 \Lambda K (the join of K and a point) is called the cone
over K and denoted cone(K).
3. Let S 0 be disjoint union of two vertices. Then S 0 \Lambda K is called the suspension
over K and denoted \SigmaK .
The geometric realisation of cone(K) (of \SigmaK ) is the topological cone (suspension)
over jKj.
The barycentric subdivision of an abstract simplicial complex K is the simplicial
complex bs(K) on the set fI; I 2 Kg of simplices of K such that fI 1 ; : : : ; I r g 2
bs(K) if and only if I 1 ae I 2 ae \Delta \Delta \Delta ae I r (after possible re­ordering). The barycen­
tre of a (polytopal) simplex \Delta n 2 R n with vertices v 1 ; : : : ; vn+1 is the point
bc(\Delta n ) = 1
n+1 (v 1 + \Delta \Delta \Delta + v n+1 ) 2 \Delta n . The barycentric subdivision of a polyhedron
P is the polyhedron bs(P) defined as follows. The vertices of bs(P) are barycen­
tres of simplices of P of all dimensions. The set of vertices fbc(\Delta i 1
1 ); : : : ; bc(\Delta i r
r )g
spans a simplex of bs(P) if and only if \Delta i 1
1 ae : : : ae \Delta i r
r in P. Obviously, one has
j bs(K)j = bs(jKj) for any abstract simplicial complex K.
Example 1.2.7 (order complex of a poset). Let (S; OE) be any poset. Define K(S)
to be the set of all chains x 1 OE x 2 OE \Delta \Delta \Delta OE x k , x i 2 S. It is easy to see that K(S) is
a simplicial complex. This complex is called the order complex of the poset (S; OE).
In particular, if (S; /) is the face poset (face lattice) of a simplicial complex K,
then K(S) is the barycentric subdivision of K.
The missing face of simplicial complex K on the set [m] is a subset I ae [m] such
that I =
2 K, but every proper subset of I is a simplex of K. A flag complex is a
simplicial complex for which every missing face has two elements. Order complexes
of posets (in particular, barycentric subdivisions) are flag complexes due to the
transitivity relation.
For any subset I ae [m] denote by K I the subcomplex of K consisting of all
simplices J 2 K such that J ae I. The link of a simplex I of simplicial com­
plex K is the subcomplex link K I ae K consisting of all simplices J 2 K such
that I [ J 2 K and I `` J = ;. For any vertex fig 2 K the cone over link K fig
(with vertex fig) is naturally a subcomplex of K, which is called the star of fig

14 VICTOR M. BUCHSTABER AND TARAS E. PANOV
and denoted star K fig. The polyhedron j star K figj consists of all (polytopal) sim­
plices of jKj that contain fig. When it is clear from the context which K is under
consideration we write link I and starfig instead of link K I and star K fig respec­
tively. Define core[m] = fi 2 [m] : starfig 6= Kg. The core of K is the subcomplex
core K = K core[m] . Thus, the core is the maximal subcomplex containing all vertices
whose stars do not coincide with K.
Example 1.2.8. 1. link(;) = K.
2. Let K be the simplex on four vertices 1; 2; 3; 4, and I = f1; 2g. Then link(I)
is a subcomplex consisting of two vertices 3 and 4.
3. Let K be the cone over K 0 with vertex p. Then linkfpg = K 0 , starfpg = K,
and core K ae K 0 .
A simplicial n­sphere is a simplicial complex homeomorphic to n­sphere S n .
(Here and below by saying ``simplicial complex K is homeomorphic to X'' we
mean that the geometric realisation jKj is homeomorphic to X.) A PL (piecewise­
linear) sphere is a simplicial sphere which is piecewise­linear homeomorphic to the
boundary of a simplex. The boundary of a simplicial n­polytope is an (n \Gamma 1)­
dimensional PL sphere. PL spheres that can be obtained in such way are called
polytopal spheres. Hence, we have the following inclusions of classes of combinatorial
objects:
polytopal spheres ae PL spheres ae simplicial spheres.
In dimension 2 any simplicial sphere is polytopal (see e.g. [89, Theorem 5.8]). How­
ever, in higher dimensions both above inclusions are strict. First examples of non­
polytopal PL 3­spheres were found by Gr¨unbaum, and the smallest such sphere
has 8 vertices. Good description of these examples can be found in [9]. A non­PL
simplicial sphere is presented in Example 1.2.10 below.
The f­vector and the h­vector of an (n \Gamma 1)­dimensional simplicial complex
K n\Gamma1 are defined in the same way as for simplicial polytopes: f (K n\Gamma1 ) =
(f 0 ; f 1 ; : : : ; fn\Gamma1 ), where f i is the number of i­dimensional simplices of K n\Gamma1 ,
and h(K n\Gamma1 ) = (h 0 ; h 1 ; : : : ; hn ), where h i are determined by (7). Here we also
assume f \Gamma1 = 1. If K n\Gamma1 = @P \Lambda is the boundary of a simplicial n­polytope P \Lambda ,
then one obviously has f (K n\Gamma1 ) = f (P \Lambda ).
Since the f­vector of a polytopal sphere coincides with the f­vector of the corre­
sponding (simplicial) polytope, the g­theorem (Theorem 1.1.17) holds for polytopal
spheres. So, it is natural to ask whether the g­theorem extends to simplicial spheres.
This question was posed by McMullen [58] as an extension of his conjecture for sim­
plicial polytopes. After 1980, when the proof of McMullen's conjecture for simplicial
polytopes was found by Billera, Lee, and Stanley, the following became perhaps the
main open combinatorial­geometrical problem concerning the f­vectors of simplicial
complexes.
Problem 1.2.9 (g­conjecture for simplicial spheres). Does the g­theorem (Theo­
rem 1.1.17) hold for simplicial spheres?
The g­conjecture is open even for PL spheres. We note that only the necessity
of g­theorem (i.e. that the g­vector is an M­vector) should be verified for simplicial
spheres. If correct, the g­conjecture would imply a characterisation of f­vectors of
simplicial spheres.
The first part of Theorem 1.1.17 (the Dehn--Sommerville equations) is known to
be true for simplicial spheres (see Corollary 1.4.15 below). The first inequality h 0 Ÿ

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 15
h 1 from the second part of g­theorem is equivalent to 1 Ÿ f 0 \Gamma n, which is obvious.
The inequality h 1 Ÿ h 2 (n – 4) is equivalent to the lower bound f 1 – nf 0 \Gamma
\Gamma n+1
2
\Delta
for the number of edges, which is also known for simplicial spheres (see [10], in fact,
the proof of the lower bound for the number of edges of simplicial polytopes can be
adapted for simplicial spheres). All these facts together imply that the g­conjecture
is true for simplicial spheres of dimension Ÿ 4. The inequality h 2 Ÿ h 3 (n – 6) from
the GLBT (the second part of Theorem 1.1.17) is open. A lot of attempts to prove
the g­conjecture were made during the last two decades. Though unsuccessful, these
attempts resulted in some very interesting reformulations of the g­conjecture. We
just mention the results of Pachner [66], [67] reducing the g­conjecture (for PL­
spheres) to some properties of bistellar moves, and the results of [84] showing that
the conjecture follows from the skeletal r­rigidity of simplicial (n \Gamma 1)­sphere for
r Ÿ
\Theta n
2
\Lambda . It was shown independently by Kalai and Stanley [80, Corollary 2.4] that
the GLBT holds for the boundary of an n­dimensional 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 g­conjecture motivated Bj¨orner and Lutz to launch the computer­aided seek for
counter examples [15]. Though their computer program, BISTELLAR, allowed to
obtain many remarkable results on triangulations of manifolds, no counter examples
to g­conjecture were found. For more history of g­theorem and related questions
see [81], [82], [89, Lecture 8].
A simplicial complex K is called a simplicial manifold (or triangulated manifold)
if the polyhedron jKj is a topological manifold. All manifolds considered in this
review are compact, connected and closed (unless otherwise stated). If K q is a
simplicial manifold, then link(I) has the homology of a (q \Gamma #I)­sphere for each non­
empty simplex I 2 K q (see Theorem 1.2.11 below). A q­dimensional PL manifold
(or combinatorial manifold) is a simplicial complex K q such that link(I) is a PL
sphere of dimension (q \Gamma #I) for each non­empty simplex I 2 K q .
Remark. If K q is a PL manifold, then for each vertex fig 2 K q the PL (q \Gamma 1)­
sphere linkfig bounds the open neighbourhood U i which is PL­homeomorphic to a
q­ball. Since any point of jK q j is contained in U i for some i, this defines a PL­atlas
on jK q j.
Example 1.2.10 (non­PL simplicial 5­sphere). Let S 3
H be any homology, but not
topological, 3­sphere, i.e. a non­simply­connected manifold with the same homology
as the ordinary 3­sphere S 3 . The Poincar'e sphere \Sigma = SO(3)=A 5 provides an
example of such a manifold. By Cannon's theorem [27], the double suspension
\Sigma 2 S 3
H is homeomorphic to S 5 . However, \Sigma 2 S 3
H can not be PL, since S 3
H appears
as the link of some 1­simplex in \Sigma 2 S 3
H .
The following theorem gives a combinatorial characterisation of simplicial com­
plexes which are simplicial manifolds of dimension – 5 and generalises the men­
tioned above result by Cannon.
Theorem 1.2.11 (Edwards [38]). For q – 5 the polyhedron of a simplicial complex
K q is a topological q­manifold if and only if link I has the homology of a (q \Gamma #I)­
sphere for each non­empty simplex I 2 K q and linkfig is simply connected for each
vertex fig 2 K.
Which topological manifolds can be triangulated is the question of great impor­
tance for combinatorial topologists. Any smooth manifold can be triangulated by

16 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Whitney's theorem. All topological 2­ and 3­dimensional manifolds can be triangu­
lated as well (for 2­manifolds this is almost obvious, for 3­manifolds see [60]). More­
over, since the link of a vertex in a simplicial 3­sphere is a 2­sphere (and 2­sphere
is always PL), all 2­ and 3­manifolds are PL. However, in dimension 4 there exist
topological manifolds that do not admit a PL­triangulation (e.g. Freedman's fake
CP 2 ). Moreover, there exist topological 4­manifolds that do not admit any triangu­
lation (e.g. Freedman's topological 4­manifold with the intersection form E 8 ). Both
examples can be found in [3]. In dimensions – 5 we have the famous combinatorial­
topological problem:
Problem 1.2.12 (Triangulation Conjecture). Is it true that every topological man­
ifold of dimension – 5 can be triangulated?
Another well­known problem of PL­topology concerns the uniqueness of a PL
structure on the topological sphere.
Problem 1.2.13. Is a PL manifold homeomorphic to the topological 4­sphere nec­
essarily a PL sphere?
Four is only dimension where the uniqueness of a PL structure for the topological
sphere is open. For dimensions Ÿ 3 the uniqueness was proved by Moise [60], and
for dimensions – 5 it follows from the work of Kirby and Siebenmann [51]. In
dimension 4 the category of PL manifolds is equivalent to the smooth category,
hence, the above problem is equivalent to if there exists an exotic 4­sphere.
More information about recent developments and open problems in combinato­
rial and PL topology can be found in [61], [72].
1.3. Simplicial complexes: commutative algebra. The commutative algebra
can be applied to combinatorics of simplicial complexes and related objects. The
main tool for translating combinatorial results and problems to the algebraic lan­
guage is the Stanley--Reisner ring of simplicial complex. This approach was outlined
by R. Stanley at the beginning of 1970's.
Remember that k[v 1 ; : : : ; vm ] denotes the graded polynomial algebra over a com­
mutative ring k with unit, deg(v i ) = 2.
Definition 1.3.1. The face ring (or the Stanley--Reisner ring) of a simplicial com­
plex K with m vertices is the quotient ring
k(K) = k[v 1 ; : : : ; vm ]=I K ;
where IK is the homogeneous ideal generated by all square­free monomials
v i 1 v i 2
\Delta \Delta \Delta v i s , i 1 ! \Delta \Delta \Delta ! i s , such that I = fi 1 ; : : : ; i s g is not a simplex of K.
Note that the ideal IK has basis consisting of square­free monomials v i 1 v i 2
\Delta \Delta \Delta v i s
such that I = fi 1 ; : : : ; i s g is a missing face of K. Ideals in the polynomial ring that
admit a basis of momomials are called monomial .
Proposition 1.3.2. Any square­free monomial ideal of the polynomial ring has the
form IK for some simplicial complex K.
Proof. For any subset I = fi 1 ; : : : ; i k g ae [m] denote by v I the square­free monomial
v i 1
\Delta \Delta \Delta v i k . Let I be a square­free monomial ideal. Set
K = fI ae [m] : v [m]nI 2 Ig:
Then one easily checks that K is a simplicial complex and I = IK .

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 17
Let P be a simple n­polytope, and P \Lambda its polar simplicial polytope. Denote
by KP the boundary of P \Lambda . Then KP is a polytopal simplicial (n \Gamma 1)­sphere.
Obviously, the face ring of P from Definition 1.1.20 coincides with that of KP from
Definition 1.3.1: k(P ) = k(KP ).
Let M = M 0 \Phi M 1 \Phi : : : be a graded k­module. The series
F (M ; t) =
1
X
i=0
(dim k M i )t i
is called the Poincar'e series of M .
Remark. In the algebraic literature the series F (M ; t) is known as the Hilbert series
or Hilbert--Poincar'e series.
The following lemma establishes the connection between two combinatorial in­
variants of a simplicial complex: the face ring and the f­vector (or the h­vector).
Lemma 1.3.3 (Stanley [81, Theorem II.1.4]). The Poincar'e series of k(K n\Gamma1 )
can be calculated as
F
\Gamma k(K n\Gamma1 ); t
\Delta =
n\Gamma1 X
i=\Gamma1
f i t 2(i+1)
(1 \Gamma t 2 ) i+1 = h 0 + h 1 t 2 + \Delta \Delta \Delta + hn t 2n
(1 \Gamma t 2 ) n ;
where (f 0 ; : : : ; fn\Gamma1 ) is the f­vector and (h 0 ; : : : ; hn ) is the h­vector of K n\Gamma1 .
Note that the second identity from Lemma 1.3.3 is an obvious corollary of (8).
Example 1.3.4. 1. Let K = \Delta n (the n­simplex). Then f i =
\Gamma n+1
i+1
\Delta for \Gamma1 Ÿ i Ÿ n,
h 0 = 1, and h i = 0 for i ? 0. Since any subset of [n + 1] is a simplex of \Delta n ,
one has k(\Delta n ) = k[v 1 ; : : : ; v n+1 ]. Then F (k(\Delta n ); t) = 1
(1\Gammat 2 ) n+1 , which agrees with
Lemma 1.3.3.
2. Let K be the boundary of an n­simplex. Then h i = 1, i = 0; 1; : : : ; n, and
k(K) = k[v 1 ; : : : ; v n+1 ]=(v 1 v 2 \Delta \Delta \Delta vn+1 ). Hence,
F
\Gamma k(K); t
\Delta = 1 + t 2 + \Delta \Delta \Delta + t 2n
(1 \Gamma t 2 ) n :
Now suppose k is a field. Let A be a graded algebra over k. The Krull dimension
of A (denoted KdA) is the maximal number of algebraically independent elements
of A. A sequence ` 1 ; : : : ; ` n of n = KdA homogeneous elements of A is called a
hsop (homogeneous system of parameters) if the Krull dimension of the quotient
A=(` 1 ; : : : ; ` n ) is zero. Equivalently, ` 1 ; : : : ; ` n is a hsop if n = KdA and A is a
finitely­generated k[` 1 ; : : : ; ` n]­module. The elements of a hsop are algebraically
independent.
Lemma 1.3.5 (Noether normalisation lemma). For any finitely­generated graded
algebra A there exists a hsop. If k is of zero characteristic and A is generated by
degree­two elements, then one can choose a degree­two hsop.
Below in this section we assume that k is of zero characteristic. 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 Ÿ 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.

18 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Remark. Regular sequences can be also defined in non­finitely­generated graded
algebras and in algebras over any integral domain. Regular sequences in the graded
polynomial ring R[a 1 ; a 2 ; : : : ; ], deg a i = \Gamma2i, on infinitely many generators, where
R is a subring of the field Q of rationals, are used in the algebraic topology for
constructing complex cobordism theories with coefficients, see [54].
Any two maximal regular sequences have the same length, which is called the
depth of A and denoted depth A. Obviously, depth A Ÿ KdA.
Definition 1.3.6. Algebra A is called Cohen--Macaulay if it admits a regular se­
quence ` 1 ; : : : ; ` n of length n = KdA.
Any regular sequence ` 1 ; : : : ; ` n of length n = KdA is a hsop. It follows that
A is Cohen--Macaulay if and only if there exists a hsop ` 1 ; : : : ; ` n such that A
is a finitely­generated free k[` 1 ; : : : ; ` n ]­module. If in addition A is generated by
degree­two elements, then one can choose ` 1 ; : : : ; ` n to be of degree two. In this
case for the Poincar'e series of A holds
F (A; t) = F
\Gamma A=(` 1 ; : : : ; ` n ); t
\Delta
(1 \Gamma t 2 ) n ;
where F (A=(` 1 ; : : : ; ` n ); t) = h 0 + h 1 t 2 + \Delta \Delta \Delta is a polynomial. The finite vector
(h 0 ; h 1 ; : : : ) is called the h­vector of A.
A simplicial complex K n\Gamma1 is called Cohen--Macaulay (over k) if its face ring
k(K) is Cohen--Macaulay. Obviously, Kdk(K) = n. Lemma 1.3.3 shows that the
h­vector of k(K) coincides with the h­vector of K.
Theorem 1.3.7 (Stanley). If K n\Gamma1 is a Cohen--Macaulay simplicial complex, then
h(K n\Gamma1 ) = (h 0 ; : : : ; hn ) is an M­vector (see section 1.2).
Proof. Let ` 1 ; : : : ; ` n be a regular sequence of degree­two elements of k(K). Then
A = k(K)=(` 1 ; : : : ; ` n ) is a graded algebra generated by degree­two elements, and
dim k A 2i = h i . Now the result follows from Theorem 1.1.18.
The following fundamental theorem characterises Cohen--Macaulay complexes.
Theorem 1.3.8 (Reisner [73]). A simplicial complex K is Cohen--Macaulay over
k if and only if for any simplex I 2 K (including I = ;) and i ! dim(link I),
e
H i (link I; k) = 0. (Here e
H i ( \Delta ; k) denotes the i­th reduced homology group with
coefficients in k.)
In particular, a simplicial sphere is a Cohen--Macaulay complex. Then Theo­
rem 1.3.7 shows that the h­vector of a simplicial sphere is an M­vector. This fact
allowed Stanley to extend the UBT (Theorem 1.1.19) to simplicial spheres.
Corollary 1.3.9 (Stanley). The Upper Bound Theorem holds for simplicial spheres.
That is, if (h 0 ; h 1 ; : : : ; hn ) is the h­vector of a simplicial (n \Gamma 1)­sphere K n\Gamma1 with
m vertices, then
h i (K n\Gamma1 ) Ÿ
\Gamma m\Gamman+i\Gamma1
i
\Delta
; 0 Ÿ i !
\Theta n
2
\Lambda
:
Proof. Since h(K n\Gamma1 ) is an M­vector, there exists a graded algebra A = A 0 \Phi
A 2 \Phi \Delta \Delta \Delta \Phi A 2n generated by degree­two elements such that dim k A 2i = h i (Theo­
rem 1.1.18). In particular, dim k A 2 = h 1 = m \Gamma n. Since A is generated by A 2 , the
number h i can not exceed the total number of monomials of degree i in (m \Gamma n)
variables. The latter is exactly
\Gamma m\Gamman+i\Gamma1
i
\Delta
.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 19
1.4. Homological properties of face rings (Stanley--Reisner rings). We
start with reviewing some homology algebra. All modules in this section are as­
sumed to be finitely­generated graded k[v 1 ; : : : ; v m ]­modules, deg v i = 2, unless
otherwise stated.
A finite free resolution of a module M is an exact sequence
0 ! R \Gammah d
\Gamma! R \Gammah+1 d
\Gamma! \Delta \Delta \Delta \Gamma! R \Gamma1 d
\Gamma! R 0 d
\Gamma! M ! 0;
(14)
where the R \Gammai are finitely­generated free modules and the maps d are degree­
preserving. The minimal number h for which a free resolution (14) exists is called
the homological dimension of M and denoted hd M . By the Hilbert syzygy the­
orem, hd M Ÿ m. A resolution (14) determines the free bigraded differential
module [R; d], where R =
L R \Gammai;j , R \Gammai;j := (R \Gammai ) j (the j­th graded compo­
nent of the free module R \Gammai ). The cohomology of [R; d] is zero in non­zero di­
mensions, while H 0 [R; d] = M . Conversely, a free bigraded differential module
[R =
L
i;j–0 R \Gammai;j ; d : R \Gammai;j ! R \Gammai+1;j ] with H 0 [R; d] = M and H \Gammai [R; d] = 0 for
i ? 0 defines a free resolution (14) with R \Gammai := R \Gammai;\Lambda =
L
j R \Gammai;j .
Remark. For reasons specified below we numerate the terms of a free resolution by
non­positive numbers, thus making it a cochain complex.
The Poincar'e series of M can be calculated from any free resolution (14).
Theorem 1.4.1. Suppose that the free k[v 1 ; : : : ; vm ]­module R \Gammai in (14) is gener­
ated by elements of degrees d 1i ; : : : ; d q i i , where q i = rankR \Gammai , i = 1; : : : ; h. Then
F (M ; t) = (1 \Gamma t 2 ) \Gammam
h
X
i=0
(\Gamma1) i (t d1i + \Delta \Delta \Delta + t dq i i ):
(15)
Example 1.4.2 (minimal resolution). For different reasons it is convenient to have
a resolution (14) for which each term R \Gammai has the smallest possible rank. The
following definition is taken from Adams' paper [2]. Let M , M 0 be two modules.
Set J (M ) = v 1 M + v 2 M + \Delta \Delta \Delta + vmM ae M . A map f : M !M 0 is called minimal
if Ker f ae J (M ). A resolution (14) is called minimal if all maps d are minimal.
Then it is easy to see that each R \Gammai has the smallest possible rank.
A minimal resolution can be constructed as follows. Take a minimal set of gen­
erators a 1 ; : : : ; a k0 for M and define R 0 to be the free k[v 1 ; : : : ; vm ]­module with
k 0 generators. Then take a minimal set of generators a 1 ; : : : ; a k1 in the kernel of
natural epimorphism R 0 !M and define R \Gamma1 to be the free k[v 1 ; : : : ; vm ]­module
with k 1 generators, and so on. On the i­th step we take a minimal set of generators
in the kernel of the previously constructed map d : R \Gammai+1 ! R \Gammai+2 and define
R \Gammai to be the free module with the corresponding generators. Note that a minimal
resolution is unique up to an isomorphism.
Example 1.4.3 (Koszul resolution). Let M = k. The k[v 1 ; : : : ; vm ]­module struc­
ture on k is defined by the map k[v 1 ; : : : ; v m ] ! k that sends each v i to 0. Let
\Lambda[u 1 ; : : : ; um ] denote the exterior algebra on m generators. The tensor product
R = \Lambda[u 1 ; : : : ; um
]\Omega k[v 1 ; : : : ; vm ] (here and
below\Omega denotes\Omega k ) becomes a dif­
ferential bigraded algebra by setting
bideg u i = (\Gamma1; 2); bideg v i = (0; 2);
du i = v i ; dv i = 0;
(16)

20 VICTOR M. BUCHSTABER AND TARAS E. PANOV
and requiring that d be a derivation of algebras. It can be shown (see [56, x7.2])
that H \Gammai [R; d] = 0 for i ? 0 and H 0 [R; d] = k. Since \Lambda[u 1 ; : : : ; um
]\Omega k[v 1 ; : : : ; v m ]
is a free k[v 1 ; : : : ; v m ]­module, it determines a free resolution of k. This resolution
is called the Koszul resolution. More precisely, it has the following form
0 ! \Lambda m [u 1 ; : : : ; um
]\Omega k[v 1 ; : : : ; v m ] \Gamma! \Delta \Delta \Delta
\Gamma! \Lambda 1 [u 1 ; : : : ; um
]\Omega k[v 1 ; : : : ; vm ] \Gamma! k[v 1 ; : : : ; vm ] \Gamma! k ! 0;
where \Lambda i [u 1 ; : : : ; um ] is the submodule of \Lambda[u 1 ; : : : ; um ] spanned by the monomials
of length i. Thus, in notations of (14) we have R \Gammai = \Lambda i [u 1 ; : : : ; um
]\Omega k[v 1 ; : : : ; vm ].
Let N be another module; then applying the
functor\Omega k[v1 ;::: ;vm ] N to (14) we
obtain the following cochain complex of graded modules:
0 \Gamma! R
\Gammah\Omega k[v1 ;::: ;vm ] N \Gamma! \Delta \Delta \Delta \Gamma! R
0\Omega k[v1 ;::: ;vm ] N \Gamma! 0
and the corresponding bigraded differential module
[R\Omega N; d]. The (\Gammai)­th coho­
mology module of the above cochain complex is denoted Tor \Gammai
k[v1 ;::: ;vm ] (M; N ), i.e.
Tor \Gammai
k[v1 ;::: ;vm ] (M; N ) := H \Gammai
[R\Omega k[v1 ;::: ;vm ] N; d]
= Ker[d : R
\Gammai\Omega k[v1 ;::: ;vm ] N ! R
\Gammai+1\Omega k[v1 ;::: ;vm ] N ]
d(R
\Gammai\Gamma1\Omega k[v1 ;::: ;vm ] N ) :
Since R \Gammai and N are graded modules, one has
Tor \Gammai
k[v1 ;::: ;vm ] (M; N ) =
M
j
Tor \Gammai;j
k[v1 ;::: ;vm ] (M; N );
where
Tor \Gammai;j
k[v1 ;::: ;vm ] (M; N ) =
Ker
\Theta
d : (R
\Gammai\Omega k[v1 ;::: ;vm ] N ) j ! (R
\Gammai+1\Omega k[v1 ;::: ;vm ] N ) j
\Lambda
d(R
\Gammai\Gamma1\Omega k[v1 ;::: ;vm ] N ) j :
Thus, we have a bigraded k[v 1 ; : : : ; vm ]­module
Tor k[v1 ;::: ;vm ] (M; N ) =
M
i;j
Tor \Gammai;j
k[v1 ;::: ;vm ] (M; N ):
The following properties of Tor \Gammai
k[v1 ;::: ;vm ] (M; N ) are well known (see e.g. [56]).
Proposition 1.4.4. (a) The module Tor \Gammai
k[v1 ;::: ;vm ] (M; N ) does not depend, up to
isomorphism, on the choice of resolution (14).
(b) Both Tor \Gammai
k[v1 ;::: ;vm ] ( \Delta ; N ) and Tor \Gammai
k[v1 ;::: ;vm ] (M; \Delta ) are covariant functors.
(c) Tor 0
k[v1 ;::: ;vm ] (M; N ) ¸ =
M\Omega k[v1 ;::: ;vm ] N .
(d) Tor \Gammai
k[v1 ;::: ;vm ] (M;N ) ¸ = Tor \Gammai
k[v1 ;::: ;vm ] (N; M ).
One can also define the A­modules TorA (M; N ) for any finitely­generated graded
commutative algebra A and (finitely­generated graded) A­modules M , N . Though
an A­free resolution (14) of M may fail to exist, there always exists a projective
resolution of M , which allows to define TorA (M; N ) in the same way as above. Note
that projective modules over polynomial algebra are free. This was known as the
Serre problem, now solved by Quillen and Suslin. However, for graded modules this
fact is not hard to prove. In our paper modules TorA (M; N ) for algebras A different
from the polynomial ring appear only in sections 4.1 and 5.3.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 21
Now let K n\Gamma1 be a simplicial complex on m vertices, M = k(K) and N = k.
Since deg v i = 2, we have
Tor k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta
=
m
M
i;j=0
Tor \Gammai;2j
k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta
(i.e. non­zero elements of Tor k[v1 ;::: ;vm ] (k(K); k) have even second degree). Define
the bigraded Betti numbers of k(K) as
fi \Gammai;2j \Gamma k(K)
\Delta := dim k Tor \Gammai;2j
k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta ; 0 Ÿ i; j Ÿ m:
(17)
Suppose that (14) is a minimal free resolution of M = k(K). Then R 0 ¸ =
k[v 1 ; : : : ; vm ] is a free k[v 1 ; : : : ; vm ]­module with one generator of degree 0. The
basis of R \Gamma1 (the minimal generator set for Ker[k[v 1 ; : : : ; vm ] ! k(K)]) consists of
elements v i 1 ;::: ;i k , deg v i 1 ;::: ;i k = 2k, such that fi 1 ; : : : ; i k g is a missing face of K.
The map d : R \Gamma1 ! R 0 takes v i 1 ;::: ;i k to v i 1 \Delta \Delta \Delta v i k . Since the maps d in (14) are
minimal, all differentials in the cochain complex
0 \Gamma! R
\Gammah\Omega k[v1 ;::: ;vm ] k \Gamma! \Delta \Delta \Delta \Gamma! R
0\Omega k[v1 ;::: ;vm ] k \Gamma! 0
are trivial. Hence, for the minimal resolution of k(K) holds
Tor \Gammai
k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta ¸ = R
\Gammai\Omega k[v1 ;::: ;vm ] k;
(18)
fi \Gammai;2j \Gamma k(K)
\Delta = rank R \Gammai;2j (= dim k[v1 ;::: ;vm ] R \Gammai;2j ):
The Betti numbers fi \Gammai;2j (k(K)) are important combinatorial invariants of the
simplicial complex K. Some results describing these numbers were obtained in [81].
The following important theorem (which was proved by combinatorial methods)
reduces the calculation of numbers fi \Gammai;2j (k(K)) to calculating the homology groups
of subcomplexes of K.
Theorem 1.4.5 (Hochster [46]). The following formula holds for the Poincar'e se­
ries of the module Tor \Gammai
k[v1 ;::: ;vm ] (k(K); k):
X
j
fi \Gammai;2j
\Gamma k(K)
\Delta
t 2j =
X
Iae[m]
\Gamma
dim k
e
H#I \Gammai\Gamma1 (K I )
\Delta
t 2(#I) ;
where K I is the subcomplex of K consisting of all simplices with vertices in I.
We mention that calculations using this theorem become very involved even for
small K. In chapter 4 we show that the numbers fi \Gammai;2j (k(K)) equal the bigraded
Betti numbers of the moment­angle complex ZK associated with the simplicial com­
plex K. This provides the alternative (topological) way for calculating the numbers
fi \Gammai;2j (k(K)).
Now we look at the Koszul resolution (Example 1.4.3).
Lemma 1.4.6. For any module M holds
Tor k[v1 ;::: ;vm ] (M; k) ¸ = H
\Theta
\Lambda[u 1 ; : : : ; um
]\Omega M; d
\Lambda
;
where H [\Lambda[u 1 ; : : : ; um
]\Omega M; d] is the cohomology of the bigraded differential module
\Lambda[u 1 ; : : : ; um
]\Omega M and d is defined as in (16).

22 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Proof. Using the Koszul resolution [\Lambda[u 1 ; : : : ; um
]\Omega k[v 1 ; : : : ; vm ]; d] for the defini­
tion of Tor k[v1 ;::: ;vm ] (k; M ) we calculate
Tor k[v1 ;::: ;vm ] (M; k) ¸ = Tor k[v1 ;::: ;vm ] (k; M )
= H
\Theta \Lambda[u 1 ; : : : ; um
]\Omega k[v 1 ; : : : ; vm
]\Omega k[v1 ;::: ;vm ] M
\Lambda ¸ = H
\Theta \Lambda[u 1 ; : : : ; um
]\Omega M
\Lambda :
Suppose now that the k[v 1 ; : : : ; v m ]­module M is a k­algebra. Then the coho­
mology of [\Lambda[u 1 ; : : : ; um
]\Omega M; d] is an algebra as well. Lemma 1.4.6 allows to invest
Tor k[v1 ;::: ;vm ] (M; k) with the canonical structure of a finite­dimensional bigraded k­
algebra. We refer to this bigraded algebra as the Tor­algebra of M . The Tor­algebra
of a simplicial complex K is the Tor­algebra of the face ring k(K).
Remark. For general N 6= k the module Tor k[v1 ;::: ;vm ] (M;N ) has no canonical
structure of an algebra even if both M and N are algebras.
Construction 1.4.7 (multigraded structure in the Tor­algebra). Invest the poly­
nomial ring k[v 1 ; : : : ; v m ] with the multigraded (more precisely, N m ­graded) struc­
ture by setting mdeg v i = (0; : : : ; 0; 2; 0; : : : ; 0), where 2 stands at the i­th place.
Then the multidegree of monomial v i 1
1 \Delta \Delta \Delta v i m
m is (2i 1 ; : : : ; 2i m ). Suppose that the
algebra M is the quotient of the polynomial ring by a monomial ideal. Then the
multigraded structure descends to M and to the terms of resolution (14). We may
assume that the differentials in the resolution preserve the multigraded structures.
Then the module Tor k[v1 ;::: ;vm ] (M; N ) acquires the canonical N \Phi N m ­grading, i.e.
Tor k[v1 ;::: ;vm ] (M;k) =
M
i–0;j 2N m
Tor \Gammai;j
k[v1 ;::: ;vm ] (M; k):
In particular, the Tor­algebra of K can be canonically made an N \Phi N m ­graded
algebra.
Remark. According to our agreement the first degree in the Tor­algebra is non­
positive. (Recall that we numerate the terms of Koszul k[v 1 ; : : : ; vm ]­free res­
olution of k by non­positive integers.) In such notations the Koszul complex
[M\Omega \Lambda[u 1 ; : : : ; um ]; d] becomes a cochain complex, and Tor k[v1 ;::: ;vm ] (M; k) is its
cohomology , not the homology as usually regarded. This is the standard trick used
for applying the Eilenberg--Moore spectral sequence, see section 4.1. It also explains
why we write Tor \Lambda;\Lambda
k[v1 ;::: ;vm ] (M; k) instead of usual Tor k[v1 ;::: ;vm ]
\Lambda;\Lambda (M; k).
The upper bound hd M Ÿ m from the Hilbert syzygy theorem can be replaced
by the following sharper result.
Theorem 1.4.8 (Auslander and Buchsbaum). hd M = m \Gamma depth M .
From now on we assume that M is generated by degree­two elements and
the k[v 1 ; : : : ; vm ]­module structure on M is defined by an epimorphism p :
k[v 1 ; : : : ; vm ] !M (both assumptions are satisfied by definition when M = k(K)
for some K). Suppose that ` 1 ; : : : ; ` k is a regular sequence of degree­two ele­
ments of M . Let J := (` 1 ; : : : ; ` k ) ae M be the ideal generated by ` 1 ; : : : ; ` k .
Choose degree­two elements t i ae k[v 1 ; : : : ; v m ] such that p(t i ) = ` i , i = 1; : : : ; k.
The ideal in k[v 1 ; : : : ; v m ] generated by t 1 ; : : : ; t k will be also denoted J . Then
k[v 1 ; : : : ; vm ]=J ¸ = k[w 1 ; : : : ; wm\Gammak ]. Under these assumptions we have the follow­
ing reduction lemma.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 23
Lemma 1.4.9. The following isomorphism of algebras holds:
Tor k[v1 ;::: ;vm ] (M; k) = Tor k[v1 ;::: ;vm ]=J (M=J ; k):
In order to prove the lemma we need the following fact from the homology
algebra.
Theorem 1.4.10 ([28, p.349]). Let \Lambda be an algebra, \Gamma its subalgebra,
and\Omega = \Lambda=\Gamma
the quotient algebra. Suppose that \Lambda is a free \Gamma­module and we are given
an\Omega ­
module A and a \Lambda­module C. Then there exists a spectral sequence fE r ; d r g such
that
E r ) Tor \Lambda (A; C); E 2 =
Tor\Omega
\Gamma
A; Tor \Gamma (C; k)
\Delta
:
Proof of Lemma 1.4.9. Set \Lambda = k[v 1 ; : : : ; vm ], \Gamma = k[t 1 ; : : : ; t k ], A = k, C = M .
Then \Lambda is a free \Gamma­module
and\Omega = \Lambda=\Gamma = k[v 1 ; : : : ; vm ]=J . Therefore, Theo­
rem 1.4.10, gives a spectral sequence
E r ) Tor k[v1 ;::: ;vm ] (M; k); E 2 =
Tor\Omega
\Gamma Tor \Gamma (M; k); k
\Delta :
Since ` 1 ; : : : ; ` k is a regular sequence, M is a free \Gamma­module. Therefore,
Tor \Gamma (M; k) =
M\Omega \Gamma k = M=J and Tor q
\Gamma (M; k) = 0 for q 6= 0:
It follows that E p;q
2 = 0 for q 6= 0. Thus, the spectral sequence collapses at the E 2
term, and we have
Tor k[v1 ;::: ;vm ] (M; k) =
Tor\Omega
\Gamma Tor \Gamma (M; k); k
\Delta = Tor k[v1 ;::: ;vm ]=J (M=J ; k);
which concludes the proof.
It follows from Lemma 1.4.9 that if M is Cohen--Macaulay of Krull dimension n,
then depth M = n, hd M = m \Gamma n, and Tor \Gammai
k[v1 ;::: ;vm ] (M; k) = 0 for i ? m \Gamma n.
Definition 1.4.11. Suppose M is a Cohen--Macaulay algebra of Krull dimension n.
Then M is called a Gorenstein algebra if Tor \Gamma(m\Gamman)
k[v1 ;::: ;vm ] (M; k) ¸ = k.
Following Stanley [81], 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 section 1.2). The following theorem characterises Gorenstein*
simplicial complexes.
Theorem 1.4.12 ([81, xII.5]). A simplicial complex K is Gorenstein* over k if
and only if for any simplex I 2 K (including I = ;) the subcomplex link I has the
homology of a sphere of dimension dim(link I).
In particular, simplicial spheres and simplicial homology spheres (simplicial man­
ifolds with the homology of a sphere) are Gorenstein* complexes. Note, however,
that a Gorenstein* complex is not necessarily a simplicial manifold (links of vertices
are not necessarily simply connected, compare with Theorem 1.2.11).
Theorem 1.4.13 ([81, xII.5]). Suppose K n\Gamma1 is a Gorenstein* complex. Then the
following identities hold for the Poincar'e series of Tor \Gammai
k[v1 ;::: ;vm ] (k(K); k), 0 Ÿ i Ÿ
m \Gamma n:
F
i
Tor \Gammai
k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta ; t
j
= t 2m F
i
Tor \Gamma(m\Gamman)+i
k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta ; 1
t
j
:
Corollary 1.4.14. If K n\Gamma1 is Gorenstein* then
F
\Gamma k(K); t
\Delta
= (\Gamma1) n F
\Gamma k(K); 1
t
\Delta
:

24 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Proof. We apply Theorem 1.4.1 to a minimal resolution of k(K). It follows from (18)
that the numerators of the summands in the right hand side of (15) are exactly
F
\Gamma
Tor \Gammai
k[v1 ;::: ;vm ] (k(K); k); t
\Delta
, i = 1; : : : ; m \Gamma n. Hence,
F
\Gamma k(K); t
\Delta = (1 \Gamma t 2 ) \Gammam
m\Gamman X
i=0
(\Gamma1) i F
i
Tor \Gammai
k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta ; t
j
:
Using Theorem 1.4.13 we get
F
\Gamma k(K); t
\Delta = (1 \Gamma t 2 ) \Gammam
m\Gamman X
i=0
(\Gamma1) i t 2m F
i
Tor \Gamma(m\Gamman)+i
k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta ; 1
t
j
=
\Gamma 1 \Gamma ( 1
t ) 2
\Delta \Gammam (\Gamma1) m
m\Gamman X
j=0
(\Gamma1) m\Gamman\Gammaj F
i
Tor \Gammaj
k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta ; 1
t
j
= (\Gamma1) n F
\Gamma k(K); 1
t
\Delta :
Corollary 1.4.15. The Dehn--Sommerville relations h i = hn\Gammai , 0 Ÿ i Ÿ n, hold
for any Gorenstein* complex K n\Gamma1 (in particular, for any simplicial sphere).
Proof. This follows from Lemma 1.3.3 and Corollary 1.4.14.
As it was pointed out by Stanley in [80], Gorenstein* complexes are the most
general objects appropriate for the generalisation of the g­theorem. (As we have
seen, polytopal spheres, PL spheres, simplicial spheres and simplicial homology
spheres are particular cases of Gorenstein* complexes).
The Dehn--Sommerville equations can be generalized even beyond Gorenstein*
complexes. In [52] Klee reproved the Dehn--Sommerville equations in the form (10)
in the more general context of Eulerian manifolds. In particular, this implies that
equations (10) (with the exception of k = 0) hold for any simplicial manifold K
of dimension n \Gamma 1. (In the case k = 0 equation (10) expresses that ü(K n\Gamma1 ) =
ü(S n\Gamma1 ).) Analogues of equations (10) were obtained by Bayer and Billera [12] (for
Eulerian posets) and Chen and Yan [30] (for arbitrary polyhedra).
In section 4.5 we obtain (by topological methods) the following form of the
Dehn--Sommerville equations for simplicial manifolds:
hn\Gammai \Gamma h i = (\Gamma1) i
\Gamma
ü(K n\Gamma1 ) \Gamma ü(S n\Gamma1 )
\Delta\Gamma n
i
\Delta
; i = 0; 1; : : : ; n:
where ü(K n\Gamma1 ) = f 0 \Gamma f 1 + : : : + (\Gamma1) n\Gamma1 fn\Gamma1 = 1 + (\Gamma1) n\Gamma1 hn is the Euler
characteristic of K n\Gamma1 and ü(S n\Gamma1 ) = 1 + (\Gamma1) n\Gamma1 is that of sphere. Note that if
K is a simplicial sphere or has odd dimension, then the above equations reduce to
the classical hn\Gammai = h i .
1.5. Cubical complexes and cubical maps. Define a q­dimensional combina­
torial­geometrical cube as a polytope combinatorially equivalent to the standard
q­cube (3). In this section cubes are combinatorial­geometrical cubes.
Definition 1.5.1. A cubical complex is a subset C ae R n represented as a union of
cubes of any dimensions in such a way that the intersection of any two cubes is a
face of each.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 25
\Delta
\Delta
\Delta
\Phi \Phi \Phi H H H
@
@
@
@
@
@
@ @
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Phi
\Phi
\Phi
H
H
H
A
A
A
\Delta \Delta
\Phi \Phi H H H
@ @
\Gamma
\Gamma
\Phi
\Phi
\Phi
H
H
A
A
XXX
@
@ @
C
C
C
\Delta
\Delta
\Delta \Delta
\Delta
\Delta \Gamma
\Gamma
\Gamma
@
@
@
A
A
A
A
A
A
\Lambda
\Lambda
\Lambda
\Gamma
\Gamma \Gamma
¸¸¸
Figure 1. Cubical complex not embedable into cubical lattice
Remark. The above definition of cubical complex is similar to Definition 1.2.2 of
geometrical simplicial complex. One could also define an abstract cubical complex ,
however this definition is more subtle (and we do not need it).
A face of a cubical complex C is a face of a cube from C. The dimension of C is
the maximal dimension of its faces. The f­vector of a cubical complex C is defined
in the standard way (f i is the number of i­faces). Some problems concerning the
f­vectors of cubical complexes are discussed in [82].
Obviously, the standard cube I q (together with all its faces) is a q­dimensional
cubical complex, which we also denote I q . Any face of I q has the form
C IaeJ = f(y 1 ; : : : ; y q ) 2 I q : y i = 0 for i 2 I; y i = 1 for i =
2 Jg;
(19)
were I ae J are two (possibly empty) subsets of [q]. We denote C J := C ;aeJ .
Unlike simplicial complexes (which are always subcomplexes of a simplex), not
any cubical complex can be realised as a subcomplex of some I q . One example of
a cubical complex not embedable as a subcomplex in any I q is shown on Figure 1.
Moreover, this complex is not embedable into the standard cubical lattice in R q
(for any q). The authors are thankful to M. I. Shtogrin for giving this example.
There is the following problem.
Problem 1.5.2 (S. P. Novikov). Characterise k­dimensional cubical complexes C
(in particular, cubical manifolds) which admit
(a) a (cubical) embedding into the standard cubical lattice in R q ;
(b) a map to the standard cubical lattice in R q whose restriction to every k­
dimensional cube identifies it with a certain k­face of the lattice.
In the case when C is homeomorphic to S 2 the above problem was solved in [37].
Problem 1.5.2 is an extension of the following problem, formulated in [37].
Problem 1.5.3 (S. P. Novikov). Suppose we are given a 2­dimensional cubical
mod 2 cycle ff in the standard cubical lattice in R 3 . Describe all maps of cubical
subdivisions of 2­dimensional surfaces onto ff such that no two different squares are
mapped to the same square of ff.
As it was told to the authors by S. P. Novikov, problem 1.5.3 have arisen in
connection with the 3­dimensional Ising model during the discussions with the
well­known physicist A.M. Polyakov.
Below we introduce some special cubical complexes, which will play a pivotal r“ole
in our theory of moment­angle complexes. Each of these cubical complexes admits

26 VICTOR M. BUCHSTABER AND TARAS E. PANOV
a canonical cubical embedding into the standard cube. We note that the problem of
embedability into the cubical lattice is closely connected with that of embedability
into the standard cube. For instance, as it was shown in [37], if a cubical subdivision
of a 2­dimensional surface is embedable into the standard cubical lattice in R q , then
it also admits a cubical embedding into I q .
Construction 1.5.4 (canonical simplicial subdivision of I m ). Let \Delta m\Gamma1 be the
simplex on the set [m], i.e. \Delta m\Gamma1 is the collection of all subsets of [m]. Assign
to each subset I = fi 1 ; : : : ; i k g ae [m] the vertex v I := C IaeI of I m . That is,
v I = ('' 1 ; : : : ; '' m ), where '' i = 0 if i 2 I and '' i = 1 otherwise. Regarding I
as a vertex of the barycentric subdivision of \Delta m\Gamma1 , we can extend the mapping
I 7! v I to the piecewise linear embedding i c of the polyhedron j bs(\Delta m\Gamma1 )j into the
(boundary complex of) standard cube I m . Under this embedding, the vertices of
j\Delta m\Gamma1 j are mapped to the vertices (1; : : : ; 1; 0; 1; : : : ; 1) 2 I m , while the barycentre
of j\Delta m\Gamma1 j is mapped to the vertex (0; : : : ; 0) 2 I m . The image i c (j bs(\Delta m\Gamma1 )j) is
the union of m facets of I m meeting at the vertex (0; : : : ; 0). For any pair I, J of
non­empty subsets of [m] such that I ae J all simplices of bs(\Delta m\Gamma1 ) of the form
I = I 1 ae I 2 ae \Delta \Delta \Delta ae I k = J are mapped to the same face C IaeJ ae I m (see (19)).
The map i c : j bs(\Delta m\Gamma1 )j ! I m extends to j cone(bs(\Delta m\Gamma1 ))j by taking the vertex
of the cone to (1; : : : ; 1) 2 I m . The resulting map is denoted by cone(i c ). Its image
is the whole I m . Hence cone(i c ) : j cone(bs(\Delta m\Gamma1 ))j ! I m is a PL homeomorphism
linear on the simplices of j cone(bs(\Delta m\Gamma1 ))j. This defines the canonical triangula­
tion of I m . Thus, the canonical triangulation of I m arises from the identification
of I m with the cone over the barycentric subdivision of \Delta m\Gamma1 .
Construction 1.5.5 (cubical subdivision of a simple polytope). Let P n ae R n be
a simple polytope with m facets F n\Gamma1
1 ; : : : ; F n\Gamma1
m . Choose a point in the relative
interior of every face of P n (including the vertices and the polytope itself). We get
the set S of 1 + f 0 + f 1 + : : : + fn\Gamma1 points (here f (P n ) = (f 0 ; f 1 ; : : : ; fn\Gamma1 ) is the
f­vector of P n ). For each vertex v 2 P n define the subset S v ae S consisting of
the points chosen inside the faces containing v. Since P n is simple, the number of
k­faces meeting at v is
\Gamma n
k
\Delta , 0 Ÿ k Ÿ n. Hence, #S v = 2 n . The set S v is the vertex
set of an n­cube, which we denote C n
v . The faces of C n
v can be described as follows.
Let G k
1 and G l
2 be two faces of P n such that v 2 G k
1 ae G l
2 . Then there are exactly
2 l\Gammak faces G of P n such that G k
1 ae G ae G l
2 . The corresponding 2 l\Gammak points from S
form the vertex set of an (l \Gamma k)­face of C n
v . We denote this face C l\Gammak
G1 aeG2 . Every face
of C n
v is C i
G1 aeG2 for some G 1 ; G 2 containing v. The intersection of any two cubes
C n
v , C n
v 0 is a face of each. Indeed, let G i ae P n be the smallest face containing both
vertices v and v 0 . Then C n
v `` C n
v 0 = C n\Gammai
G i aeP n is the face of both I n
v and I n
v 0 . Thus, we
have constructed a cubical subdivision of P n with fn\Gamma1 (P n ) cubes of dimension n.
We denote this cubical complex C(P n ).
There is an embedding of C(P n ) to I m constructed as follows. Every (n \Gamma k)­
face of P n is the intersection of k facets: G n\Gammak = F n\Gamma1
i 1
`` : : : `` F n\Gamma1
i k . We map
the corresponding point of S to the vertex ('' 1 ; : : : ; '' m ) 2 I m , where '' i = 0 if
i 2 fi 1 ; : : : ; i k g and '' i = 1 otherwise. This defines 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
(Construction 1.5.4) we extend this mapping to the PL embedding i P : P n ! I m .
For each vertex v = F n\Gamma1
i 1
`` \Delta \Delta \Delta `` F n\Gamma1
i n
2 P n we have
i P (C n
v ) =
\Phi (y 1 ; : : : ; ym ) 2 I m : y j = 1 for j =
2 fi 1 ; : : : ; i ng
\Psi ;
(20)

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 27
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta \DeltaA
A
A
A
A
A
A
A
A
A
A A
H H H
\Phi
\Phi
\Phi
­
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma \Gamma
\Gamma \Gamma
A F E
G
B D
C
P n
i P
A F
B
G
C D
E
I m
Figure 2. The embedding i P : P n ! I m for n = 2, m = 3.
i.e. i P (C n
v ) = C fi1 ;::: ;i ng ae I m (in the notations of (19)). The embedding i P : P n !
I m for n = 2, m = 3 is shown on Figure 2.
We summarise the facts from the above construction in the following statement
Theorem 1.5.6. A simple polytope P n with m facets can be split into cubes C n
v ,
one for each vertex v 2 P n . The resulting cubical complex C(P n ) embeds canonically
into the boundary of I m , as described by (20).
Lemma 1.5.7. The number of k­faces of the cubical complex C(P n ) is given by
f k
\Gamma C(P n )
\Delta
=
n\Gammak X
i=0
\Gamma n\Gammai
k
\Delta
fn\Gammai\Gamma1 (P n )
=
\Gamma n
k
\Delta
fn\Gamma1 (P n ) +
\Gamma n\Gamma1
k
\Delta
fn\Gamma2 (P n ) + \Delta \Delta \Delta + f k\Gamma1 (P n ); k = 0; : : : ; n:
Proof. This follows from the fact that the k­faces of C(P n ) are in one­to­one corre­
spondence with pairs G i
1 ; G i+k
2 of faces of P n such that G i
1 ae G i+k
2 .
Construction 1.5.8. Let K n\Gamma1 be a simplicial complex on [m]. Then K is natu­
rally a subcomplex of \Delta m\Gamma1 , and bs(K) is a subcomplex of bs(\Delta m\Gamma1 ). As it follows
from Construction 1.5.4, there is a PL embedding i c j bs(K) : j bs(K)j ! I m . The
image i c (j bs(K)j) is an (n \Gamma 1)­dimensional cubical subcomplex of I m , which we
denote cub(K). Then
cub(K) =
[
;6=IaeJ2K
C IaeJ ae I m ;
(21)
i.e. cub(K) is the union of faces C IaeJ ae I m over all pairs I ae J of non­empty
simplices of K.
Construction 1.5.9. Since cone(bs(K)) is a subcomplex of cone(bs(\Delta m\Gamma1 )), Con­
struction 1.5.4 also provides a PL embedding cone(i c )j cone(bs(K)) : j cone(bs(K))j !
I m . The image of this embedding is an n­dimensional cubical subcomplex of I m ,

28 VICTOR M. BUCHSTABER AND TARAS E. PANOV
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
u
u
u
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
0
(a) K = : \Delta
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
u
u
u
u
u
u
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
0
(b) K = @ \Delta 2
Figure 3. The cubical complex cub(K).
which we denote cc(K). Then one easily obtains that
cc(K) =
[
J2K
C IaeJ ae I m :
(22)
Since C IaeJ ae C ;aeJ = C J , we also can write cc(K) =
S
J2K C J .
The following statement summarises the results of two previous constructions.
Theorem 1.5.10. For any simplicial complex K on the set [m] there is a PL
embedding of the polyhedron jKj into I m linear on the simplices of bs(K). The image
of this embedding is the cubical subcomplex (21). Moreover, there is a PL embedding
of the polyhedron j cone(K)j into I m linear on the simplices of cone(bs(K)). The
image of this embedding is the cubical subcomplex (22).
As in the case of simplicial complexes, a cubical complex C 0 is called a cubical
subdivision of cubical complex C if each cube of C is a union of finitely many cubes
of C 0 .
Proposition 1.5.11. For every cubical subcomplex C there exists a cubical subdi­
vision C 0 that can be realised as a subcomplex of some I q .
Proof. Subdividing each cube of C as described in Construction 1.5.4 we obtain a
simplicial complex, say K C . Then applying Construction 1.5.8 to K C we get a cubical
complex that subdivides jK C j = C and embeds into some I q (see Theorem 1.5.10)
as the subcomplex cub(K C ).
Example 1.5.12. Figure 3 (a) shows the cubical complex cub(K) for the case
when K is the disjoint union of 3 vertices (K = : \Delta). Figure 3 (b) shows that for
the case when K is the boundary complex of a 2­simplex (K = @ \Delta 2 ). The cubical
complexes cc(K) in both cases are indicated on Figure 4 (a) and (b).
Remark. As a topological space, cub(K) is homeomorphic to jKj, while cc(K) is
homeomorphic to j cone(K)j. For simplicial complex cone(K) one may construct the
cubical complex cub(cone(K)), which is also homeomorphic to j cone(K)j. How­
ever, as cubical complexes, cc(K) and cub(cone(K)) differ (since cone(bs(K)) 6=
bs(cone(K))).
Let P be a simple n­polytope and KP the corresponding simplicial (n \Gamma 1)­sphere
(the boundary of the polar simplicial polytope P \Lambda ). Then cc(KP ) coincides with the

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 29
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
u
u
u
u
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
0
(a) K = : \Delta
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
u
u
u
u
u
u
u
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
0
(b) K = @ \Delta 2
Figure 4. The cubical complex cc(K).
cubical complex C(P ) from Construction 1.5.5 (more precisely, cc(KP ) = i P (C(P ))).
Thus, the Construction 1.5.5 is a particular case of Construction 1.5.9.
Remark. Some versions of our previous constructions already appeared in the lit­
erature. A version of Construction 1.5.9 can be found in [33, p. 434] (it was used
there for studying some torus actions; we return to this in the next chapter). A
version of our cubical subcomplex cub(K) ae I m appeared in [75] in connection
with Problem 1.5.2.
2. Toric and quasitoric manifolds
2.1. Toric varieties. Toric varieties appeared in algebraic geometry in the be­
ginning of 1970's in connection with compactification problems for algebraic tori
actions (see below). Very quickly the geometry of toric varieties became one of
the most fascinating topics in algebraic geometry and found applications in many
mathematical sciences, which otherwise seem far from algebraic geometry. We have
already mentioned the proof of necessity of g­theorem for simplicial polytopes given
by Stanley. Other remarkable applications include counting lattice points and vol­
umes of lattice polytopes; relations with Newton polytopes and singularities (after
Khovanskii and Kushnirenko); discriminants, resultants and hypergeometric func­
tions (after Gelfand, Kapranov and Zelevinsky); reflexive polytopes and mirror
symmetry for Calabi--Yau toric hypersurfaces (after Batyrev). The standard refer­
ences in the toric geometry are Danilov's survey [32] and books by Fulton [40] and
Oda [62]. More recent survey article by Cox [31] covers new applications, including
mentioned above. We are not going to give a new survey 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 g­theorem.
Let C \Lambda = C n f0g denote the multiplicative group of complex numbers. The
product (C \Lambda ) n of n copies of C \Lambda 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 \Lambda ) n as the algebraic torus. The torus T n is a subgroup of
the algebraic torus (C \Lambda ) n in the standard way:
T n =
\Phi\Gamma e 2úi'1 ; : : : ; e 2úi'n
\Delta 2 C n \Psi ;
(23)
where (' 1 ; : : : ; 'n ) varies over R n .

30 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Definition 2.1.1. A toric variety is a normal algebraic variety M containing the
algebraic torus (C \Lambda ) n as a Zariski open subset in such a way that the natural action
of (C \Lambda ) n on itself extends to an action on M .
Hence, (C \Lambda ) n acts on M with a dense orbit.
Every toric variety is encoded by a set of combinatorial data, namely by a (ra­
tional polyhedral) fan in some R n .
Let R n be the Euclidean space and Z n ae R n the integral lattice. Given a finite
set of vectors l 1 ; : : : ; l s 2 R n , define the convex polyhedral cone oe spanned by
l 1 ; : : : ; l s as
oe = fr 1 l 1 + \Delta \Delta \Delta + r s l s 2 R n : r i – 0g:
Any convex polyhedral cone is a convex polyhedron in the sense of Definition 1.1.2.
Hence, the faces of a convex polyhedral cone are defined. A cone oe is rational if
its generator vectors l 1 ; : : : ; l s can be taken from Z n and is strongly convex if it
contains no line through the origin. All cones considered below are strongly convex
and rational. A cone is called simplicial (respectively non­singular) if it is generated
by a part of a basis of R n (respectively Z n ). A fan is a set \Sigma of cones in some R n
such that each face of a cone in \Sigma is also a cone in \Sigma, and the intersection of two
cones in \Sigma is a face of each. A fan \Sigma is called simplicial (respectively non­singular)
if all cones of \Sigma are simplicial (respectively non­singular). A fan \Sigma in R n is called
complete if the union of all cones from \Sigma is R n .
A fan \Sigma in R n determines the toric variety M \Sigma of complex dimension n, which
orbit structure is described by the combinatorics of \Sigma. That is, the k­dimensional
cones of \Sigma correspond to the codimension­k orbits of the algebraic torus action
on M 2n . In particular, the n­dimensional cones correspond to the fixed points,
while the origin corresponds to the unique dense orbit. The toric variety M \Sigma is
compact if and only if \Sigma is complete. If \Sigma is simplicial then M \Sigma is an orbifold (i.e.
locally homeomorphic to the quotient of R 2n by a finite group action). Finally, if \Sigma
is non­singular, then, as one should expect, M \Sigma is non­singular (smooth). Smooth
toric varieties sometimes are called toric manifolds.
Let \Sigma be a simplicial fan in R n with m one­dimensional cones (or rays). Choose
generator vectors l 1 ; : : : ; l m for these m rays to be integer and primitive (i.e. with
relatively prime integer coordinates). The fan \Sigma defines the simplicial complex K \Sigma
on the vertex set [m], which is called the underlying complex of \Sigma. By definition,
fi 1 ; : : : ; i k g ae [m] is a simplex of K \Sigma if l i 1 ; : : : ; l i k span a cone of \Sigma. It is easy to
see that if \Sigma is complete, then K \Sigma is a simplicial (n \Gamma 1)­sphere.
Denote l ij := (l j ) i , 1 Ÿ i Ÿ n, 1 Ÿ j Ÿ m. Here l j = (l 1j ; : : : ; l nj ) t 2 Z n . Assign
to each vector l j the indeterminate v j of degree 2, and define linear forms
` i := l i1 v 1 + \Delta \Delta \Delta + l im vm 2 Z[v 1 ; : : : ; vm ]; 1 Ÿ i Ÿ n:
Denote by J \Sigma the ideal in Z[v 1 ; : : : ; vm ] spanned by these linear forms, i.e. J \Sigma =
(` 1 ; : : : ; ` n ). The images of ` 1 ; : : : ; ` n and J \Sigma in the Stanley--Reisner ring Z(K \Sigma ) =
Z[v 1 ; : : : ; vm ]=I K \Sigma (see Definition 1.3.1) will be denoted by the same symbols
` 1 ; : : : ; ` n and J \Sigma .
Theorem 2.1.2 (Danilov and Jurkiewicz). Let \Sigma be a complete non­singular fan
in R n , and M \Sigma the corresponding toric variety. Then
(a) The Betti numbers (the ranks of homology groups) of M \Sigma vanish in odd
dimensions, while in even dimensions are given by
b 2i (M \Sigma ) = h i (K \Sigma ); i = 0; 1; : : : ; n;

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 31
where h(K \Sigma ) = (h 0 ; : : : ; hn ) is the h­vector of K \Sigma .
(b) The closures of orbits corresponding to the one­dimensional cones of \Sigma are
codimension­2 submanifolds (divisors) D i of M \Sigma . Let v i , 1 Ÿ i Ÿ m, be the cor­
responding 2­dimensional cohomology classes. Then the cohomology ring of M \Sigma is
given by
H \Lambda (M \Sigma ; Z) ¸ = Z[v 1 ; : : : ; vm ]=(I K \Sigma + J \Sigma ) = Z(K \Sigma )=J \Sigma :
Moreover, ` 1 ; : : : ; ` n is a regular sequence in Z(K \Sigma ).
This theorem was proved by Jurkiewicz for projective smooth toric varieties and
by Danilov [32, Theorem 10.8] in the general case. Note that the first part of
Theorem 2.1.2 follows from the second part and Lemma 1.3.3.
Theorem 2.1.2 shows that the cohomology of M \Sigma is identified with the Chow
ring [40, x 5.1] of M \Sigma and is generated by two­dimensional classes. Note that the
ideal IK \Sigma
depends only on simplicial complex K \Sigma (i.e. on the intersection lattice
of fan), while J \Sigma depends on fan \Sigma itself.
Remark. As it was shown by Danilov, Theorem 2.1.2 remains true for simplicial
fans and corresponding toric varieties if one replaces the coefficient ring Zby any
field of zero characteristic (e.g. Q).
Construction 2.1.3 (Normal fan and toric varieties arising from polytopes). Suppose
we are given an n­polytope (1) with vertices in the integer lattice Z n ae R n . (Such
polytopes are called integral (or lattice).) Then the vectors l i in (1), 1 Ÿ i Ÿ m,
can be chosen integer and primitive, and the numbers a i can be chosen integer.
Note that each l i is the normal vector of the facet F i ae P n and l i is pointing inside
the polytope P . Define the complete fan \Sigma(P ) whose cones are generated by sets
of normal vectors l i 1 ; : : : ; l i k such that the corresponding facets F i 1 ; : : : ; F i k have
non­empty intersection in P . The fan \Sigma(P ) is called the normal fan of P . Alter­
natively, if 0 2 P then the normal fan consists of cones over the faces of the polar
polytope P \Lambda . Define the toric variety MP := M \Sigma(P ) . The variety MP is smooth if
and only if P is simple and the normal vectors l i 1 ; : : : ; l i n of any set of n facets
F i 1 ; : : : ; F i n meeting at the same vertex form a basis of Z n .
Remark. Any combinatorial simple polytope is rational , that is, admits a geomet­
rical realisation with rational (or, equivalently, integer) vertex coordinates. Indeed,
there is a small perturbation of defining inequalities in (1) that makes all of them
rational but does not change the combinatorial type (since the half­spaces defined
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
realisation with integral vertex coordinates one should take the magnified polytope
kP 0 for appropriate k 2 Z. Quite surprising thing is that there exist non­rational
convex polytopes (non­simple and non­simplicial), see [89, Example 6.21]. Return­
ing to simple polytopes, we note that different realisations of a given combinatorial
simple polytope as lattice polytopes may produce different (even topologically)
toric varieties MP . At the same time there exist combinatorial simple polytopes
that do not admit any geometrical realisation with smooth MP . We discuss one
such example in the next section (see Example 2.2.14).
In the rest of this section all polytopes are assumed to be simple. Construc­
tion 2.1.3 allows to define the simplicial fan \Sigma(P ) and the toric variety MP from
any lattice simple polytope P . However, the polytope P contains more information

32 VICTOR M. BUCHSTABER AND TARAS E. PANOV
than the fan \Sigma(P ). Indeed, besides the normal vectors l i , we also have numbers
a i 2 Z, 1 Ÿ i Ÿ m, (see (1)). The linear combination D = a 1 D 1 + \Delta \Delta \Delta + amDm (see
Theorem 2.1.2) is an ample divisor . It defines a projective embedding MP ae CP r
for some r (which can be taken to be the number of vertices of P ). Thus, all toric
varieties from polytopes are projective. Conversely, given a smooth projective toric
variety M ae CP r , one gets a 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
! := a 1 v 1 + \Delta \Delta \Delta + am v m 2 H 2 (MP ; Q) be the cohomology class of D.
Theorem 2.1.4 (Hard Lefschetz theorem for toric varieties). Let P n be a lattice
simple polytope (1), MP the toric variety defined by P , and ! = a 1 v 1 + \Delta \Delta \Delta +am v m 2
H 2 (MP ; Q). Then the maps
H n\Gammai (MP ; Q) \Delta ! i
\Gamma\Gamma\Gamma\Gamma! H n+i (MP ; Q); 1 Ÿ i Ÿ n;
are isomorphisms.
It follows from the projectivity that if MP is smooth then it is K¨ahler, and ! is the
class of the K¨ahler 2­form.
Remark. If one replaces the ordinary cohomology by the intersection cohomology ,
then Theorem 2.1.4 holds for any projective toric variety, not necessarily arising
from a simple lattice polytope (see the discussion in [40, x 5.2]).
Example 2.1.5. The complex projective space CP n = f(z 0 : z 1 : \Delta \Delta \Delta : z n ); z i 2 Cg
is a toric variety. (C \Lambda ) n acts on CP n by (t 1 ; : : : ; t n ) \Delta (z 0 : z 1 : \Delta \Delta \Delta : z n ) =
(z 0 : t 1 z 1 : \Delta \Delta \Delta : t n z n ). Obviously, (C \Lambda ) n ae C n ae CP n is a dense open sub­
set. The fan defining CP n consists of cones generated by all proper subsets
of (n + 1) vectors e 1 ; : : : ; e n ; \Gammae 1 \Gamma \Delta \Delta \Delta \Gamma e n in R n . Theorem 2.1.2 identifies
the cohomology ring H \Lambda (C P n ; Z) = Z[u]=(u n+1 ), dimu = 2, with the quotient
Z[v 1 ; : : : ; vn+1 ]=(v 1 \Delta \Delta \Delta v n+1 ; v 1 \Gamma v n+1 ; : : : ; vn \Gamma vn+1 ). The toric variety CP n arises
from a polytope: CP n = MP , where P is the standard n­simplex (2). The class
! 2 H 2 (C P n ; Q) from Theorem 2.1.4 in this case is ! = v n+1 .
Now we are ready to give Stanley's argument for the necessity of g­theorem for
simple polytopes.
Proof of necessity of Theorem 1.1.17. Realise the simple polytope as a lattice poly­
tope P n ae R n . Let MP be the corresponding toric variety. Part (a) is already
proved (Theorem 1.1.11). It follows from Theorem 2.1.4 that the multiplica­
tion by ! 2 H 2 (MP ; Q) is a monomorphism H 2i\Gamma2 (MP ; Q) ! H 2i (MP ; Q) for
i Ÿ
\Theta n
2
\Lambda . This together with the part (b) of Theorem 2.1.2 gives h i\Gamma1 Ÿ h i ,
0 Ÿ i Ÿ
\Theta n
2
\Lambda , thus proving (b). To prove (c), define the graded commutative Q­
algebra A := H \Lambda (MP ; Q)=(!). Then A 0 = Q, A 2i = H 2i (MP ; Q)=! \Delta H 2i\Gamma2 (MP ; Q)
for 1 Ÿ i Ÿ
\Theta n
2
\Lambda
, and A is generated by degree­two elements (since so is H \Lambda (MP ; Q)).
It follows from Theorem 1.1.18 that the numbers dimA 2i = h i \Gamma h i\Gamma1 , 0 Ÿ i Ÿ
\Theta n
2
\Lambda
,
are the components of an M­vector, thus proving (c) and the whole theorem.
Remark. The Dehn--Sommerville equations now follow from the Poincar'e duality
for MP (which holds also for singular MP provided that P is simple).
Now we consider the action of the torus T n ae (C \Lambda ) n on a non­singular compact
toric variety M . This action is locally equivalent to the standard action of T n

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 33
on C n (see the next section for the precise definition). The orbit space M=T n is
homeomorphic to an n­ball, invested with the topological structure of manifold
with corners by the fixed point sets of appropriate subtori, see [40, x 4.1]. (Roughly
speaking, a manifold with corners is a space that is locally modelled by open subsets
of the positive cone R n
+ (4). From this description it is easy to deduce the strict
definition, see [49], which we omit here.)
Construction 2.1.6. Let P n be a simple polytope. For any vertex v 2 P n denote
by U v the open subset of P n obtained by deleting all faces not containing v. Ob­
viously, U v is diffeomorphic to R n
+ (and even affinely isomorphic to an open set of
R n
+ containing 0). It follows that P n is a manifold with corners, with atlas fU v g.
If M = MP arises from some (simple) polytope P n , then the orbit space M=T n
is diffeomorphic, as a manifold with corners, to P n . Furthermore, there exists an
explicit map MP ! R n (the moment map) with image P n and T n ­orbits as fibres,
see [40, x4.2]. (We consider relationships with moment maps and some aspects of
symplectic geometry in more details in section 5.2.) Under this map, the interior
of a codimension­k face of P n is identified with the set of orbits having the same
k­dimensional isotropy subgroup. In particular, the action is free over the interior of
the polytope. Regarded as a smooth manifold, MP can be identified with quotient
space T n \Theta P n =¸ for some equivalence relation ¸. Such description of the torus
action on a non­singular toric variety motivated the appearance of an important
topological analogue of toric varieties, namely, the theory of quasitoric manifolds.
2.2. Quasitoric manifolds. A quasitoric manifold is a smooth manifold with a
torus action whose properties are similar to that of the (compact) torus action on a
non­singular projective toric variety. This notion appeared in [33] under the name
``toric manifold''. We use the term ``quasitoric manifold'', since ``toric manifold'' is
occupied in the algebraic geometry for ``non­singular toric variety''. In the conse­
quent definitions we follow [33], taking into account some adjustments from [26].
As in the previous section, we regard the torus T n as the standard subgroup (23)
in C \Lambda , therefore specifying the orientation and the coordinate subgroups T i ¸ = 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 R n
+ . The canonical projection
T n \Theta R n
+ ! C n : (t 1 ; : : : ; t n ) \Theta (x 1 ; : : : ; xn ) ! (t 1 x 1 ; : : : ; t nxn )
allows to identify C n with the quotient space T n \Theta R n
+ =¸ for some equivalence
relation ¸, which will play an important r“ole in our future considerations.
Let M 2n be a 2n­dimensional manifold with an action of the torus T n . Say that
the T n ­action is locally standard if every point x 2 M 2n lies in some T n ­invariant
neighbourhood U (x) for which there exists a /­equivariant homeomorphism f :
U (x) ! W with some (T n ­stable) open subset W ae C n . That is, there is an
automorphism / : T n ! T n such that f(t \Delta y) = /(t)f(y) for all t 2 T n , y 2 U (x).
The orbit space for a locally standard action of T n on M 2n is an n­dimensional
manifold with corners. Quasitoric manifolds correspond to the important case when
this orbit space is diffeomorphic, as manifold with corners, to a simple polytope P n .
Definition 2.2.1. Given a simple polytope P n , a manifold M 2n with locally stan­
dard T n ­action is said to be a quasitoric manifold over P n if there is a projection
map ú : M 2n ! P n whose fibres are the orbits of the action.

34 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Under projection ú, points that have same isotropy subgroup of codimension k are
taken to the interior of a certain k­face of P n . In particular, the action of T n is free
over the interior of P n , while the vertices of P n correspond to the T n ­fixed points
of M 2n .
Remark. Two simple polytopes are combinatorially equivalent if and only if they
are diffeomorphic as manifolds with corners.
Suppose P n has m facets F 1 ; : : : ; Fm . For every facet F i the pre­image ú \Gamma1 (F i )
is a submanifold M 2(n\Gamma1)
i ae M 2n with isotropy subgroup a circle T (F i ) in T n . Then
T (F i ) =
\Phi \Gamma e 2úi–1i' ; : : : ; e 2úi–ni'
\Delta 2 T n
\Psi ;
(24)
where ' 2 R and –– i = (– 1i ; : : : ; – ni ) t 2 Z n is a primitive vector. This –– i is
determined by T (F i ) only up to a sign. The choice of this sign specifies an orientation
of T (F i ). For now we do not care about this sign and choose it arbitrary. More
detailed treatment of these signs is the subject of the next section. We refer to –– i
as the facet vector corresponding to F i . The action of T n =T (F i ) on M i describes it
as a quasitoric manifold over F i . The correspondence
` : F i 7! T (F i )
(25)
is called the characteristic map of M 2n . Suppose we have a codimension­k face G n\Gammak
written as the intersection of k facets: G n\Gammak = F i 1
`` \Delta \Delta \Delta ``F i k . Then the submanifolds
M i 1 ; : : : ; M i k intersect transversally in a facial submanifold M (G) 2(n\Gammak) . The map
T (F i 1 ) \Theta \Delta \Delta \Delta \Theta T (F i k ) ! T n is injective since T (F i 1 ) \Theta \Delta \Delta \Delta \Theta T (F i k ) is identified with
the k­dimensional isotropy subgroup of M (G) 2(n\Gammak) . It follows that the vectors
–– i 1 ; : : : ; –– i k form a part of integral basis of Z n .
Let \Lambda be the integer (n \Theta m)­matrix whose i­th column is formed by the coordi­
nates of the facet vector –– i , i = 1; : : : ; m. Each vertex v 2 P n is the in intersection
of n facets: v = F i 1
`` \Delta \Delta \Delta `` F i n . Let \Lambda (v) := \Lambda (i 1 ;::: ;i n ) be the maximal minor of \Lambda
formed by the columns i 1 ; : : : ; i n . Then
det \Lambda (v) = \Sigma1:
(26)
The correspondence
G n\Gammak 7! isotropy subgroup of M (G) 2(n\Gammak)
extends the characteristic map (25) to a map from the face lattice of P n to the
lattice of subtori of T n .
Like in the case of standard action of T n on C n , there is a projection T n \Theta P n !
M 2n whose fibre over x 2 M 2n has the form (isotropy subgroup of x) \Theta (orbit of x).
This argument can be used for reconstructing the quasitoric manifoldfrom any given
pair (P n ; `), where P n is a (combinatorial) simple polytope and ` is a map from
facets of P n to one­dimensional subgroups of T n such that `(F i 1 ) \Theta \Delta \Delta \Delta\Theta `(F i k ) ! T n
is injective whenever F i 1
`` \Delta \Delta \Delta `` F i k 6= ;. Such (P n ; `) is called a characteristic pair .
The map ` directly extends to a map from the face lattice of P n to the lattice of
subtori of T n .
Construction 2.2.2 (Quasitoric manifold from a characteristic pair). Note that
each point q of P n lies in the relative interior of a unique face G(q). Now construct
the identification space (T n \Theta P n )=¸, where (t 1 ; q) ¸ (t 2 ; q) if and only if t 1 t \Gamma1
2 lies
in the subtorus `(G(q)). The free action of T n on T n \Theta P n obviously descends to an
action on (T n \Theta P n )=¸ with quotient P n . The latter action is free over the interior

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 35
of P n and has a fixed point for each vertex of P n . Just as P n is covered by the open
sets U v , based on the vertices and diffeomorphic to R n
+ (see Construction 2.1.6),
so the space (T n \Theta P n )=¸ is covered by open sets (T n \Theta U v )=¸ homeomorphic to
(T n \Theta R n
+ )=¸, and therefore to C n . This implies that the T n ­action on (T n \Theta P n )=¸
is locally standard, and therefore M 2n (`) := (T n \Theta P n )=¸ is a quasitoric manifold.
Given an automorphism / : T n ! T n , say that two quasitoric manifolds M 2n
1 ,
M 2n
2 over the same P n are /­equivariantly diffeomorphic if there is a diffeomorphism
f : M 2n
1 ! M 2n
2 such that f(t \Delta x) = /(t)f(x) for all t 2 T n , x 2 M 2n
1 . The
automorphism / induces an automorphism / \Lambda of the lattice of subtori of T n . Any
such automorphism descends to a /­translation of characteristic pairs, in which the
two characteristic maps differ by / \Lambda . The following proposition is proved in [26,
Proposition 2.6] and generalises Proposition 1.8 of [33].
Proposition 2.2.3. For any automorphism /, Construction 2.2.2 defines a bi­
jection between /­equivariant diffeomorphism classes of quasitoric manifolds and
/­translations of pairs (P n ; `).
When / is the identity, we deduce that two quasitoric manifolds are equivariantly
diffeomorphic if and only if their characteristic maps are the same.
Now we are going to construct a cellular decomposition of M 2n with only even
dimensional cells and calculate the Betti numbers accordingly, following [33].
Construction 2.2.4. We recall ``Morse­theoretical arguments'' from the proof of
Dehn--Sommerville relations (Theorem 1.1.11). There we turned the 1­skeleton of
P n into a directed graph and defined the index ind(v) of a vertex v 2 P n as the
number of incident edges that point towards v. These inward edges define a face G v
of dimension ind(v). Denote by b
G v the subset of G v obtained by deleting all faces
not containing v. Obviously, b
G v is diffeomorphic to R ind(v)
+ and is contained in the
open set U v ae P n from Construction 2.1.6. Then e v := ú \Gamma1 b
G v is identified with
C ind(v) , and the union of the e v over all vertices of P n define a cell decomposition
of M 2n . Note that all cells are even­dimensional and the closure of the cell e v is
the facial submanifold M (G v ) 2 ind(v) ae M 2n . This construction was earlier used by
Khovanskii [50] for constructing cellular decompositions of toric varieties.
Proposition 2.2.5. The Betti numbers of M 2n vanish in odd dimensions, while
in even dimensions are given by
b 2i (M 2n ) = h i (P n ); i = 0; 1; : : : ; n;
where h(P n ) = (h 0 ; : : : ; hn ) is the h­vector of P n .
Proof. The 2i­th Betti number equals the number of 2i­dimensional cells in the cel­
lular decomposition constructed above. This number equals the number of vertices
of index i, which is h i (P n ) by the argument from the proof of Theorem 1.1.11.
Given a quasitoric manifold M 2n with characteristic map (25) and facet vectors
–– i = (– 1i ; : : : ; – ni ) t 2 Z n , i = 1; : : : ; m, define linear forms
` i := – i1 v 1 + \Delta \Delta \Delta + –im v m 2 Z[v 1 ; : : : ; v m ]; 1 Ÿ i Ÿ n:
(27)
The images of these linear forms in the face ring Z(P n ) will be denoted by the same
letters.

36 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Lemma 2.2.6 (Davis and Januszkiewicz). For any quasitoric manifold M 2n over P n ,
the sequence ` 1 ; : : : ; ` n is a (degree­two) regular sequence in Z(P n ).
Let J ` denote the ideal in Z(P n ) generated by ` 1 ; : : : ; ` n .
Theorem 2.2.7 (Davis and Januszkiewicz). Let v i , 1 Ÿ i Ÿ m, be the 2­dimensional
cohomology classes dual to the submanifolds M 2(n\Gamma1)
i ae M 2n . Then the cohomology
ring of M 2n is given by
H \Lambda (M 2n ; Z) ¸ = Z[v 1 ; : : : ; vm ]=(I P + J ` ) = Z(P n )=J ` :
We give proofs for the above two statements in section 3.4.
Remark. Change of sign of vector –– i corresponds to passing from v i to \Gammav i in the
description of the cohomology ring from Theorem 2.2.7. This fact will be crucial in
the next section.
Example 2.2.8. A non­singular projective toric variety MP arising from a lattice
simple polytope P n is a quasitoric manifold over P n . The corresponding charac­
teristic map ` : F i 7! T (F i ) is defined by (24), where –– i = (– 1i ; : : : ; – ni ) t are
the normal vectors l i of facets of P n , i = 1; : : : ; m (see (1)). The corresponding
characteristic n \Theta m­matrix \Lambda is the matrix L from Construction 1.1.4. In partic­
ular, if P n is the standard simplex \Delta n (2) then MP is CP n (Example 2.1.5) and
\Lambda = (E j \Gamma 1), where E is the unit n \Theta n­matrix and 1 is the column of units. See
also Example 2.4.14 below.
Generally, a non­singular toric variety is not necessarily a quasitoric manifold:
although the orbit space (for the action of T n ) is a manifold with corners (see sec­
tion 2.1), it may fail to be diffeomorphic (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 any such example. In [40, p. 71] one can find
the example of a complete non­singular fan \Sigma in R 3 which can not be obtained
by taking the cones with vertex 0 over the faces of a geometrical simplicial poly­
tope. Nevertheless, since the corresponding simplicial complex K \Sigma is a simplicial
2­sphere, it is combinatorially equivalent to a polytopal 2­sphere. This means that
the non­singular toric variety M \Sigma , although being non­projective, is still a quasitoric
manifold.
Problem 2.2.9. Give an example of a non­singular toric variety which is not a
quasitoric manifold.
This problem seems to be not very hard and reduces to constructing a complete
non­singular fan \Sigma whose associated simplicial complex K \Sigma is the Barnette sphere
or any other non­polytopal PL­sphere (see section 1.2). As it was pointed out by
N. Strickland, the definition of quasitoric manifold can be modified in such a way
that non­singular toric varieties become quasitoric manifolds. For this purpose, the
simple polytope in Definition 2.2.1 is to be replaced by the polytopal complex
dual to a simplicial sphere. The corresponding constructions are currently being
developed.
On the other hand, it is easy to construct a quasitoric manifold which is not a
toric variety. The simplest example is the manifold CP 2 #CP 2 (the connected sum
of two copies of CP 2 ). It is a quasitoric manifold over the square I 2 (this follows
from the construction of equivariant connected sum, see [33, 1.11], section 2.3 and
corollary 2.5.5 below). However, CP 2 # CP 2 do not admit even an almost complex

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 37
structure (i.e. a complex structure in the tangent bundle). The following problem
arises.
Problem 2.2.10. Let P n be simple polytope with m facets, ` a characteristic
map (25), and M 2n (`) the derived quasitoric manifold (Construction 2.2.2). Find
conditions on P n and ` so that M 2n (`) admits a T n ­invariant complex (respectively
almost complex) structure.
The almost complex case of the above problem was formulated in [33, Prob­
lem 7.6]. Example 2.2.8 (characteristic functions of lattice polytopes) provides a
sufficient condition for Problem 2.2.10 (since a non­singular toric variety is a com­
plex manifold). However, this condition is obviously non­necessary even for the
existence of a complex structure. Indeed, there are non­singular (non­projective)
toric varieties that do not arise from any lattice simple polytope (see the already
mentioned example from [40, p. 71]). At the same time, we do not know any example
of non­toric complex quasitoric manifold.
Problem 2.2.11. Find an example of a non­toric 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 on the stable tangent bundle.
The corresponding constructions are the subject of the next section.
Another class of problems arises in connection with the classification of qua­
sitoric manifolds over a given combinatorial simple polytope. The general setting of
this problem is the subject of section 2.5. Example 2.2.14 below shows that there
are combinatorial simple polytopes that do not admit a characteristic map (and
therefore can not arise as orbit spaces for quasitoric manifolds).
Problem 2.2.12. Give a combinatorial description of the class of polytopes P n
that admit a characteristic map (25).
A generalisation of this problem is given in section 4.4 (Problem 4.4.1).
A characteristic map is determined by an integer n \Theta m­matrix \Lambda which satis­
fies (26) for every vertex v 2 P n . The equation (det \Lambda (v) ) 2 = 1 defines a hyper­
surface in the space M(n;m;Z) of integer n \Theta m­matrices. Problem 2.2.12 can be
reformulated in the following way.
Proposition 2.2.13. The set of characteristic matrices coincides with the inter­
section ``
v2P n
\Phi (det \Lambda (v) ) 2 = 1
\Psi
(28)
of hypersurfaces in the space M(n;m;Z), where v varies over the vertices of the
polytope P n .
Example 2.2.14 (polytope that do not admit a characteristic function, [33, Example 1.22]).
Suppose P n is a 2­neighbourly simple polytope with m – 2 n facets (e.g. the polar of
the cyclic polytope C n (m) (Example 1.1.9) with n – 4 and m – 2 n ). Then P n does
not admit a characteristic map, and therefore can not appear as the quotient space
for a quasitoric manifold. Indeed, by Proposition 2.2.13, it is sufficient to show that
intersection (28) is empty. Since m – 2 n , any matrix \Lambda 2 M(n;m;Z) (without zero
columns) contains two columns, say i­th and j­th, which coincide modulo 2. Since

38 VICTOR M. BUCHSTABER AND TARAS E. PANOV
P n is 2­neighbourly, the corresponding facets F i and F j have non­empty intersec­
tion in P n . Hence, columns i and j of \Lambda enter some minor of the form \Lambda (v) . This
implies that the determinant of this minor is even and intersection (28) is empty.
2.3. Stably complex structures and quasitoric representatives in cobor­
dism classes. This section is the review of results obtained by N. Ray and the
first author in [25] and [26] supplied with some additional comments.
A stably complex structure on a (smooth) manifold M is defined by a complex
structure on the vector bundle Ü (M ) \Phi R k for some k, where Ü (M ) is the tangent
bundle of M and R k denotes the trivial real k­bundle over M . A stably complex
manifold (in other notations, weakly almost complex manifold or U ­manifold) is
a manifold with fixed stably complex structure, that is, a pair (M; ¸), where ¸
is a complex bundle isomorphic, as a real bundle, to Ü (M ) \Phi R k for some k. If
M itself is a complex manifold, then it possesses the canonical stably complex
structure (M; Ü (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\Omega U . By the theorem of Milnor and
Novikov,\Omega U ¸ = Z[a 1 ; a 2 ; : : : ], deg a i = 2i (see [61], [83]). The
ring\Omega U serves as
the coefficient ring for a generalised (co)homology theory known as the complex
(co)bordisms.
Stably complex manifolds were the main subject of F. Hirzebruch's talk at the
1958 International Congress of mathematicians (see [86]). Using Milnor hypersur­
faces (Example 2.3.10) and the Milnor--Novikov theorem it was shown that every
complex cobordism class contains a non­singular algebraic variety (not necessarily
connected). The following problem is still open.
Problem 2.3.1 (Hirzebruch). Which cobordism classes
in\Omega U contain connected
non­singular algebraic varieties?
Example 2.3.2. The 2­dimensional cobordism
group\Omega U
2
¸ = Zis generated by the
class of [C P 1 ] (Riemannian sphere). Every cobordism class k[C P 1 ]
2\Omega U
2 contains
a non­singular algebraic variety, namely, the disjoint union of k copies of CP 1
for k ? 0 and the disjoint union of k copies of a genus 2 Riemannian surface
for k ! 0. However, a connected algebraic variety is contained only in cobordism
classes k[CP 1 ] for k Ÿ 1.
The problem of choice of appropriate generators for the
ring\Omega U plays a pivotal
r“ole in the cobordism theory and its applications. In this section we give a solution
of the quasitoric analogue of problem 2.3.1, recently obtained in [25] and [26]. This
solution relies upon an important additional structure on a quasitoric manifold,
namely, the omniorientation, which provides a canonical stably complex structure
described in the combinatorial terms.
Let ú : M 2n ! P n be a quasitoric manifold with characteristic map `. Since the
torus T n (23) 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 specified by orienting the ambient
space R n .)
Definition 2.3.3. An omniorientation of a quasitoric manifold M 2n consists of a
choice of an orientation for M 2n and for every submanifold M 2(n\Gamma1)
i = ú \Gamma1 (F i ),
i = 1; : : : ; m.
There are therefore 2 m+1 omniorientations in all for given M 2n .

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 39
An omniorientation of M 2n determines an orientation for every normal bundle
š i := š(M i ae M 2n ), i = 1; : : : ; m. Since every š i is a real 2­plane bundle, an
orientation of š i allows to interpret it as a complex line bundle. The isotropy sub­
group T (F i ) of the submanifold M 2(n\Gamma1)
i = ú \Gamma1 (F i ) acts on the normal bundle š i ,
i = 1; : : : ; m. Thus, we have the following statement.
Proposition 2.3.4. 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 (24).
We refer to a characteristic map ` as directed if all circles `(F i ), i = 1; : : : ; m,
are oriented. This implies that signs of the facet vectors –– i = (– 1i ; : : : ; – ni ) t , i =
1; : : : ; m, are determined unambiguously. In the previous section we organised the
facet vectors into the integer n \Theta m matrix \Lambda. This matrix satisfies (26). Due to (24),
knowing a matrix \Lambda is equivalent to knowing a directed characteristic map. Let Z F
denote the m­dimensional free Z­module spanned by the set F of facets of P n .
Then \Lambda defines an epimorphism – : Z F ! Z n by –(F i ) = –– i and an epimorphism
T F ! T m , which we denote by the same letter –. In the sequel we write Z m for
Z F and T m for T F with the agreement that the vector e i of the standard basis
of Z m corresponds to F i 2 Z F , i = 1; : : : ; m (and the same for T m ). A directed
characteristic pair (P n ; \Lambda) consists of a simple polytope P n and an integer matrix
\Lambda (or, equivalently, an epimorphism – : Z m !Z n ) that satisfies (26).
Proposition 2.3.4 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 by Construction 2.2.2 is omnioriented.
Construction 2.3.5. The orientation of the bundle š i over M i defines an integral
Thom class in the cohomology group H 2 (M (š i )), represented by a complex line
bundle over the Thom complex M (š i ). We pull this back along the Pontryagin--
Thom collapse M 2n ! M (š i ), and denote the resulting bundle ae i . The restriction
of ae i to M i ae M 2n is š i .
Theorem 2.3.6 ([26, Theorem 3.8]). Every omniorientation of a quasitoric man­
ifold M 2n determines a stably complex structure on it by means of the following
isomorphism of real 2m­bundles:
Ü (M 2n ) \Phi R 2(m\Gamman) ¸ = ae 1 \Phi \Delta \Delta \Delta \Phi ae m :
It follows that a directed characteristic pair (P n ; \Lambda) determines a complex cobor­
dism class [M 2n ; ae 1 \Phi \Delta \Delta \Delta \Phi ae m ]
2\Omega U . At the same time, the above constructions
can be directly applied to computing the complex cobordism
ring\Omega \Lambda
U (M 2n ) of an
omnioriented quasitoric manifold.
Theorem 2.3.7 ([26, Proposition 5.3]). Let v i denote the first cobordism Chern
class c 1 (ae i )
2\Omega 2
U (M 2n ) of the bundle ae i , 1 Ÿ i Ÿ m. Then the cobordism ring of
M 2n is given by
\Omega \Lambda
U (M 2n ) ¸
=\Omega U [v 1 ; : : : ; v m ]=(I P + J \Lambda );
where the ideals IP and J \Lambda are defined in the same way as in Theorem 2.2.7.
Note that the Chern class c 1 (ae i ) is Poincar'e dual to the inclusion M 2(n\Gamma1)
i ae M 2n
by construction of ae i . This highlights the remarkable fact that the complex bor­
dism
groups\Omega U
\Lambda (M 2n ) are spanned by embedded submanifolds. By definition, the
fundamental cobordism class hM 2n i
2\Omega 2n
U (M 2n ) is dual to the bordism class of a

40 VICTOR M. BUCHSTABER AND TARAS E. PANOV
point. Thus, hM 2n i = v i 1 \Delta \Delta \Delta v i n for any set i 1 ; : : : ; i n such that F i 1 `` \Delta \Delta \Delta `` F i n is a
vertex of P n .
Example 2.3.8 (bounded flag manifold [24]). A bounded flag in C n+1 is a com­
plete flag U = fU 1 ae U 2 ae \Delta \Delta \Delta ae Un+1 = C n+1 g for which U k contains the
coordinate subspace C k\Gamma1 (spanned by the first k \Gamma 1 standard basis vectors) for
2 Ÿ k Ÿ n. As it is shown in [26, Example 2.8], the 2n­dimensional manifold Bn of
all bounded flags in C n+1 is a quasitoric manifold over the cube I n with respect to
the action induced by t \Delta z = (t 1 z 1 ; : : : ; t n z n ; z n+1 ) on C n+1 (here t 2 T n ).
Example 2.3.9. A family of manifolds B i;j , 0 Ÿ i Ÿ j, is introduced in [25]. The
manifold B i;j consists of pairs (U; W ), where U is a bounded flag in C i+1 (see
Example 2.3.8) and W is a line in U ?
1 \Phi C j \Gammai . So B i;j is a smooth CP j \Gamma1 ­bundle
over B i . It is shown in [26, Example 2.9] that B i;j is a quasitoric manifold over the
product I i \Theta D j \Gamma1 .
The canonical stably complex structures and omniorientations on the manifolds
Bn and B i;j are described in [26, examples 4.3, 4.5].
Remark. The product of two quasitoric manifolds M 2n1
1 and M 2n2
2 over polytopes
P n1
1 and P n2
2 is a quasitoric manifold over P n1
1 \Theta P n2
2 . This construction extends to
omnioriented quasitoric manifolds and is compatible with stably complex structures
(details can be found in [26, Proposition 4.7]).
It is shown in [25] that the cobordism classes of B i;j multiplicatively generate
the
ring\Omega U . So every 2n­dimensional complex cobordism class may be represented
by a disjoint union of products
B i 1 ;j 1
\Theta B i 2 ;j 2
\Theta \Delta \Delta \Delta \Theta B i k ;j k ;
(29)
where
P k
q=1 (i q + j q ) \Gamma 2k = n. Each such component is a quasitoric manifold,
under the product quasitoric structure. This result is the substance of [25]. The
stably complex structures of products (29) are induced by omniorientations, and
are therefore also preserved by the torus action.
Example 2.3.10. The standard set of multiplicative generators
for\Omega U consists
of projective spaces CP i , i – 0, and Milnor hypersurfaces H i;j ae CP i \Theta CP j ,
1 Ÿ i Ÿ j. The hypersurfaces H i;j are defined by
H i;j =
n
(z 0 : \Delta \Delta \Delta : z i ) \Theta (w 0 : \Delta \Delta \Delta : w j ) 2 CP i \Theta CP j :
i
X
q=0
z q w q = 0
o
:
However, as it was shown in [25], the hypersurfaces H i;j are not quasitoric manifolds
for i – 2.
To give genuinely toric representatives (which are, by definition, connected) for
each cobordism class of dimension ? 2, it remains only to replace the disjoint
union of products (29) with their connected sum. This is done in [26, x6] using
Construction 1.1.7 and its extension to omnioriented quasitoric manifolds.
Theorem 2.3.11 (Buchstaber and Ray). In dimensions ? 2, every complex cobor­
dism class contains a quasitoric manifold, necessarily connected, whose stably com­
plex structure is induced by an omniorientation, and is therefore compatible with
the action of the torus.
This theorem gives a solution to the quasitoric analogue of Problem 2.3.1.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 41
2.4. Combinatorial formulae for Hirzebruch genera of quasitoric man­
ifolds. 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 [68], [69].
Namely, using arguments similar to that from the proof of Theorem 1.1.11 we con­
struct a circle action with only isolated fixed points on any quasitoric manifold M 2n .
If M 2n is omnioriented, then this action preserves the stably complex structure and
its local representations near fixed points are retrieved from the characteristic ma­
trix \Lambda. This allows to calculate Hirzebruch's ü y ­genus as the sum of contributions
corresponding to the vertices of polytope. These contributions depend only on the
``local combinatorics'' near the vertex. After some adjustments the formula also
allows to calculate the signature and the Todd genus of M 2n .
Definition 2.4.1. The Hirzebruch genus associated with the series
Q(x) = 1 +
X
q k x k ; q k 2 Q;
is the ring homomorphism 'Q
:\Omega U ! Q that to each cobordism class [M 2n ]
2\Omega U
2n
assigns the value given by the formula
'Q [M 2n ] =
i n
Y
i=1
Q(x i ); hM 2n i
j
:
Here M 2n is a smooth manifold whose stable tangent bundle Ü (M 2n ) is a complex
bundle with complete Chern class in cohomology
c(Ü ) = 1 + c 1 (Ü ) + \Delta \Delta \Delta + c n (Ü ) =
n
Y
i=1
(1 + x i );
and hM 2n i is a fundamental class in homology.
The ü y ­genus is the Hirzebruch genus associated with the series
Q(x) = x(1 + ye \Gammax(1+y) )
1 \Gamma e \Gammax(1+y) ;
where y 2 R is a parameter. In particular cases y = \Gamma1; 0; 1 we obtain the n­th
Chern number, the Todd genus and the signature of the manifold M 2n correspond­
ingly.
Provided that M 2n is a complex manifold, the value ü y (M 2n ) can be calculated
in terms of Euler characteristics of the Dolbeault complexes on M 2n . The general
information about Hirzebruch genera can be found in [45].
In this section we assume that we are given an omnioriented quasitoric manifold
M 2n over some P n with characteristic matrix \Lambda. This specifies a stably complex
structure on M 2n , as described in the previous section. The orientation of M 2n
defines the fundamental class hM 2n i 2 H 2n (M 2n ; Z).
Construction 2.4.2. Suppose v is a vertex of P n expressed as the intersection of
n facets:
v = F i 1
`` \Delta \Delta \Delta `` F i n :
(30)
Assign to each facet F i k the edge E k :=
T
j 6=k F i j (that is, E k contains v and is
opposite to F i k ). Let e k be a vector along E k beginning at v. Then e 1 ; : : : ; e n is

42 VICTOR M. BUCHSTABER AND TARAS E. PANOV
a basis of R n , which may be either positively or negatively oriented depending on
the ordering of facets in (30). Throughout this section this ordering is assumed to
be so that e 1 ; : : : ; e n is a positively oriented basis.
Once we specified an ordering of facets in (30), the facet vectors –– i 1 ; : : : ; –– i n at
v may in turn constitute either positively or negatively oriented basis depending on
the sign of the determinant of \Lambda (v) = (–– i 1 ; : : : ; –– i n ) (see (26)).
Definition 2.4.3. The sign of a vertex v = F i 1 `` \Delta \Delta \Delta `` F i n of P n is
oe(v) := det \Lambda (v) :
The collection of signs of vertices of P n provides an important invariant of an
oriented omnioriented quasitoric manifold. Note that reversing the orientation of
M 2n changes all signs oe(v) to the opposite. At the same time changing the di­
rection of a facet vector reverses the signs only for the vertices contained in the
corresponding facet.
Let E be an edge of P n . The isotropy subgroup of the 2­dimensional submanifold
ú \Gamma1 (E) ae M 2n is an (n \Gamma 1)­dimensional subtorus, which we denote by T (E). Then
we can write
T (E) =
\Phi\Gamma
e 2úi'1 ; : : : ; e 2úi'n
\Delta 2 T n : ¯ 1 ' 1 + : : : + ¯n'n = 0
\Psi
(31)
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 (Z n ) \Lambda ; it is
determined by E only up to a sign. There is no canonical way to fix these signs
simultaneously for all edges. However, the following lemma shows that the omnior­
ientation of M 2n provides a canonical way to choose signs of edge vectors ``locally''
at each vertex.
Lemma 2.4.4. For any vertex v 2 P n , signs of edge vectors ¯¯ 1 ; : : : ; ¯¯ n meeting
at v can be chosen in such way that the n \Theta n­matrix M (v) := (¯¯ 1 ; : : : ; ¯¯ n ) satisfies
the identity
M t
(v) \Delta \Lambda (v) = E;
where E is the identity matrix. In other words, ¯¯ 1 ; : : : ; ¯¯ n and –– i 1 ; : : : ; –– i n are
conjugate bases.
Proof. At the beginning we choose signs of the edge vectors at v arbitrary, and
express v as in (30). Then ¯¯ k is the edge vector corresponding to the edge E k
opposite to F i k , k = 1; : : : ; n. It follows that E k ae F i l and T (F i l ) ae T (E k ) for
l 6= k. Hence,
h¯¯ k ; –– i l i = 0; l 6= k;
(32)
(see (24) and (31)). Since ¯¯ k is a primitive vector, it follows from (32) that
h¯¯ k ; –– i k i = \Sigma1. Changing the sign of ¯¯ i k if necessary, we obtain
h¯¯ k ; –– i k i = 1;
which together with (32) gives M t
(v) \Delta \Lambda (v) = E, as needed.
In the sequel we assume that signs of edge vectors at each v are chosen as in the
above lemma. It follows that the edge vectors ¯¯ 1 ; : : : ; ¯¯ n meeting at v constitute
an integer basis of Z n and
det M (v) = oe(v):
(33)

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 43
Suppose M 2n = MP is a smooth toric variety arising from a lattice simple
polytope P defined by (1). Then –– i = l i , i = 1; : : : ; m (see Example 2.2.8), while
the edge vectors at v 2 P n are the primitive integer vectors e 1 ; : : : ; e n along the
edges beginning at v. It follows from Construction 2.4.2 that oe(v) = 1 for any v.
Lemma 2.4.4 in this case expresses the fact that e 1 ; : : : ; e n and l i 1 ; : : : ; l i n are
conjugate bases of Z n . Similarly, the following statement holds.
Proposition 2.4.5. Suppose that the omniorientation of the quasitoric manifold
M 2n arises from a T m ­invariant almost complex structure (i.e. complex structure
in the tangent bundle Ü (M 2n )). Then oe(v) = 1 for any vertex v 2 P n .
Proof. The orientation of P n is determined by the canonical orientation of the
almost complex manifoldM 2n and the orientation of the torus (23). Since the almost
complex structure on M 2n is T n ­invariant, it induces almost complex structures on
the T (F i )­fixed submanifolds M 2(n\Gamma1)
i . It follows that for any vertex (30) the vectors
–– i 1 ; : : : ; –– i n constitute a positively oriented basis of R n .
Proposition 2.4.5 provides a necessary condition for the existence of a T n ­
invariant almost complex structure on M 2n (see Problem 2.2.10).
Remark. Globally Lemma 2.4.4 provides two directions (signs) for an edge vector,
one for each of its ends. These signs are always different provided that M 2n is a
complex manifold (e.g. a smooth toric variety), but in general this fails to be true.
Let šš = (š 1 ; : : : ; šn ) t 2 Z n be a primitive vector such that
h¯¯; šši 6= 0 for any edge vector ¯¯:
(34)
The vector šš defines the one­dimensional oriented subtorus
T š š :=
\Phi\Gamma e 2úiš1' ; : : : ; e 2úišn'
\Delta 2 T n : ' 2 R
\Psi :
Lemma 2.4.6 ([69, Theorem 2.1]). For any šš satisfying (34) the circle T š
š acts on
M 2n with only isolated fixed points (corresponding to the vertices of P n ). For each
vertex v = F i 1
`` \Delta \Delta \Delta `` F i n the action of T š
š induces a representation of S 1 in the
tangent space T v M 2n with weights h¯¯ 1 ; šši; : : : ; h¯¯ n ; šši.
Remark. If M 2n = MP is a smooth toric variety, then the genericity condition (34)
is equivalent to that from the proof of Theorem 1.1.11.
Definition 2.4.7. Suppose we are given a primitive vector šš satisfying (34). De­
fine the index of a vertex v 2 P n as the number of negative weights of the S 1 ­
representation in T v M 2n from Lemma 2.4.6. That is, if v = F i 1 `` \Delta \Delta \Delta `` F i n , then
ind š
š (v) = f#k : h¯¯ k ; šši ! 0g:
Remark. The index of a vertex v can be also defined in terms of the facet vectors
at v. Indeed, Lemma 2.4.4 shows that if v = F i 1 `` \Delta \Delta \Delta `` F i n , then
šš = h¯¯ 1 ; šši–– i 1 + \Delta \Delta \Delta + h¯¯ n ; šši–– i n :
Hence, ind š
š (v) equals the number of negative coefficients in the representation of
šš as a linear combination of basis vectors –– i 1 ; : : : ; –– i n .
Theorem 2.4.8 ([68, Theorem 6], [69, Theorem 3.1]). For any šš satisfying (34),
the ü y ­genus of M 2n can be calculated as
ü y (M 2n ) =
X
v2P n
(\Gammay) ind š (v) oe(v):

44 VICTOR M. BUCHSTABER AND TARAS E. PANOV
This theorem is proved by applying the Atiyah--Hirzebruch formula [6] to the
circle action defined in Lemma 2.4.6.
The value of the ü y ­genus ü y (M 2n ) at y = \Gamma1 equals the n­th Chern number
c n (¸)hM 2n i for any 2n­dimensional stably complex manifold [M 2n ; ¸]. In quasitoric
case Theorem 2.4.8 gives
c n [M 2n ] =
X
v2P n
oe(v):
(35)
If M 2n is a complex manifold (e.g. a smooth toric variety), then oe(v) = 1 for all
vertices v 2 P n and c n [M 2n ] equals the Euler characteristic e(M 2n ). Hence, for
complex M 2n the Euler characteristic equals the number of vertices of P n (which is
well known for toric varieties). For general quasitoric M 2n the Euler characteristic
is also equal to the number of vertices of P n (since the Euler characteristic of any
S 1 ­manifold equals the sum of Euler characteristics of fixed submanifolds), however
the latter number may differ from c n [M 2n ] (see Example 2.4.15 below).
The value of the ü y ­genus at y = 1 is the signature (or the L­genus). Theo­
rem 2.4.8 gives in this case
Corollary 2.4.9. The signature of an omnioriented quasitoric manifold M 2n can
be calculated as
sign(M 2n ) =
X
v2P n
(\Gamma1) indš (v) oe(v):
Being an invariant of an oriented cobordism class, the signature of M 2n does not
depend on a stably complex structure (i.e. on an omniorientation) and is determined
only by an orientation of M 2n (or P n ). The following modification of Corollary 2.4.9
provides a formula for sign(M 2n ) that does not depend on an omniorientation.
Corollary 2.4.10 ([69, Corollary 3.3]). The signature of an oriented quasitoric
manifold M 2n can be calculated as
sign(M 2n ) =
X
v2P n
det(e¯¯ 1 ; : : : ; e ¯¯ n );
where e
¯¯ k , k = 1; : : : ; n, are the edge vectors at v oriented in such way that
he¯¯ k ; šši ? 0.
If M 2n = MP is a smooth toric variety, then oe(v) = 1 for any v 2 P n , and
Corollary 2.4.9 gives
sign(MP ) =
X
v2P n
(\Gamma1) indš (v) :
Since in this case the index ind š (v) is the same as the index from the proof of
Theorem 1.1.11, we obtain
sign(MP ) =
n
X
k=1
(\Gamma1) k h k (P ):
(36)
Note that if n is odd, then the right hand side of the above formula vanishes due
to the Dehn--Sommerville equations. The formula (36) appears in a more general
context in recent work of Leung and Reiner [55]. The quantity in the right hand
side of (36) arises in the following well known combinatorial conjecture.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 45
Problem 2.4.11 (Charney--Davis conjecture). Let K be a (2q \Gamma 1)­dimensional
Gorenstein* flag complex with h­vector (h 0 ; h 1 ; : : : ; h 2q ). Is it true that
(\Gamma1) q (h 0 \Gamma h 1 + \Delta \Delta \Delta + h 2q ) – 0?
This conjecture was made in [29, Conjecture D] for flag simplicial homology
spheres. Stanley [82, Problem 4] proposed to extend it to Gorenstein* complexes.
The Charney--Davis conjecture is closely connected with the following differential­
geometrical conjecture.
Problem 2.4.12 (Hopf conjecture). Let M 2q be a Riemannian manifold of non­
positive sectional curvature. Is it true that the Euler characteristic ü(M 2n ) satisfies
the inequality
(\Gamma1) q ü(M 2q ) – 0?
Both above conjectures are known to be true for q = 1; 2 and for some special
cases. More details can be found in [29]. For more relations of the two problems
with the signature of a toric variety see [55].
Now we turn again to the ü y ­genus of an omnioriented quasitoric manifold. The
next important particular case is the Todd genus corresponding to y = 0. In this
case the summands in the formula from Theorem 2.4.8 are not defined for the
vertices of index 0, so it requires some additional analysis.
Theorem 2.4.13 ([68, Theorem 7], [69, Theorem 3.4]). The Todd genus of an
omnioriented quasitoric manifold can be calculated as
td(M 2n ) =
X
v2P n :ind š (v)=0
oe(v)
(the sum is taken over all 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 1.1.11). Since oe(v) = 1 for every v 2 P n ,
Theorem 2.4.13 gives td(MP ) = 1, which is well known (see e.g. [40, x5.3]; for
algebraic varieties the Todd genus equals the arithmetic genus).
If M 2n is an almost complex manifold, then td(M 2n ) ? 0 by Proposition 2.4.5
and Theorem 2.4.13.
Example 2.4.14. The projective space CP 2 , regarded as a toric variety, arises
from the standard lattice 2­simplex \Delta 2 (with vertices v 1 = (0; 0), v 2 = (1; 0),
v 3 = (0; 1)). The orientation is standard (determined by the complex structure).
The omniorientation is determined by the facet vectors –– 1 ; –– 2 ; –– 3 , which in this case
are the primitive normal vectors pointing inside the polytope. The edge vectors are
the primitive vectors along edges pointing out of the vertex. This can be seen on
Figure 5. The corresponding stably complex structure is the standard one, that is,
determined by the isomorphism of bundles Ü (C P 2 ) \Phi R 2 ¸ = ¯
j \Phi ¯
j \Phi ¯ j, where j is
the canonical Hopf line bundle. Let us calculate the Todd genus and the signature
from Corollary 2.4.9 and Theorem 2.4.13. We have oe(v 1 ) = oe(v 2 ) = oe(v 3 ) = 1.
Take šš = (1; 2), then ind š (v 1 ) = 0, ind š (v 2 ) = 1, ind š (v 3 ) = 2 (recall that the
index is the number of negative scalar products of edge vectors with šš). Thus,
sign(C P 2 ) = sign[CP 2 ; ¯ j \Phi ¯
j \Phi ¯ j] = 1, td(C P 2 ) = td[CP 2 ; ¯
j \Phi ¯ j \Phi ¯
j] = 1.

46 VICTOR M. BUCHSTABER AND TARAS E. PANOV
@
@
@
@
@
@
@
@
@
@
@
@
@
''!
/
oe
v 1 (1; 0) (\Gamma1; 0) v 2
–– 2 = (0; 1)
(0; 1)
–– 1 = (1; 0)
(0; \Gamma1)
v 3
(1; \Gamma1)
(\Gamma1; 1)
–– 3 = (\Gamma1; \Gamma1)
Figure 5. Ü (CP 2 ) \Phi C ' ¯
j \Phi ¯
j \Phi ¯ j
@
@
@
@
@
@
@
@
@
@
@
@
@
''!
/
oe
v 1 (1; 0) (1; 0) v 2
–– 2 = (0; 1)
(0; 1)
–– 1 = (1; 0)
(0; 1)
v 3
(1; \Gamma1)
(\Gamma1; 1)
–– 3 = (1; 1)
Figure 6. Ü (CP 2 ) \Phi C ' ¯
j \Phi ¯
j \Phi j
Example 2.4.15. Now consider CP 2 with the omniorientation determined by the
three facet vectors –– 1 ; –– 2 ; –– 3 shown on Figure 6. This omniorientation differs from
the previous example by the sign of –– 3 . The corresponding stably complex structure
is determined by the isomorphism Ü (C P 2 ) \Phi R 2 ¸ = ¯ j \Phi ¯
j \Phi j. Using (33) we calculate
oe(v 1 ) =
fi fi fi fi 1 0
0 1
fi fi fi fi = 1; oe(v 2 ) =
fi fi fi fi \Gamma1 1
1 0
fi fi fi fi = \Gamma1; oe(v 3 ) =
fi fi fi fi 0 1
1 \Gamma1
fi fi fi fi = \Gamma1:
For šš = (1; 2) we find ind š (v 1 ) = 0, ind š (v 2 ) = 0, ind š (v 3 ) = 1. Thus, sign[CP 2 ; ¯
j \Phi
¯
j \Phi j] = 1, td[CP 2 ; ¯ j \Phi ¯
j \Phi j] = 0. Note that in this case formula (35) gives
c n [C P 2 ; ¯ j \Phi ¯
j \Phi j] = oe(v 1 ) + oe(v 2 ) + oe(v 3 ) = \Gamma1, while the Euler number of CP 2
is c n [C P 2 ; ¯ j \Phi ¯
j \Phi ¯ j] = 3.
T n ­equivariant stably complex and almost complex manifolds were considered
in works of Hattori [44] and Masuda [57] as a separate generalisation (called the

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 47
unitary toric manifolds) of toric varieties. Instead of Davis and Januszkiewicz's
characteristic maps, Masuda in [57] uses the notion of multi­fan to describe the
combinatorial structure of the orbit space. The multi­fan is a collection of cones
which may overlap unlike a usual fan. The Todd genus of a unitary toric manifold
was calculated in [57] via the degree of the overlap of cones in the multi­fan. This
result is equivalent to our Theorem 2.4.13 in the case of quasitoric manifolds.
2.5. The classification problem for quasitoric manifolds over a given sim­
ple polytope. More precisely, there are two classification problems: the equivari­
ant (i.e. up to an equivariant diffeomorphism) and the topological (i.e. up to a
diffeomorphism). Due to Proposition 2.2.3, the equivariant classification problem
reduces to the description of all characteristic maps for given simple polytope P n .
The topological classification problem usually requires an additional analysis.
Let M 2n be a quasitoric manifold over P n with characteristic map `. We suppose
here that the first n facets F 1 ; : : : ; Fn share a common vertex.
Lemma 2.5.1. Up to /­equivalence, we may assume that `(F i ) is the i­th coordi­
nate subtorus T i ae T n , i = 1; : : : ; n.
Proof. Since the one­dimensional subtori `(F i ), i = 1; : : : ; n, generate T n , we may
define / as any automorphism of T n that maps `(F i ) to T i , i = 1; : : : ; n.
It follows that M 2n admits such an omniorientation that the corresponding char­
acteristic n \Theta m­matrix \Lambda has the form (E j \Lambda), where E is the identity matrix and
\Lambda is a certain integer n \Theta (m \Gamma n)­matrix.
In the simplest case P n = \Delta n the equivariant (and topological) classification of
quasitoric manifolds reduces to the following easy result.
Proposition 2.5.2. Any quasitoric manifold over the simplex \Delta n is equivariantly
diffeomorphic to CP n (regarded as a toric variety, see examples 2.1.5 and 2.2.8).
Proof. The characteristic map for CP n has the form
` CP n (F i ) = T i ; i = 1; : : : ; n; ` CP n (Fn+1 ) = S d ;
where S 1
d := f(e 2úi' ; : : : ; e 2úi' ) 2 T n g, ' 2 R, is the diagonal subgroup in T n . Let
M 2n be a quasitoric manifold over \Delta n with characteristic map ` M . We may assume
that ` M (F i ) = T i , i = 1; : : : ; n, by Lemma 2.5.1. Then it easily follows from (26)
that
` M (Fn+1 ) =
\Phi \Gamma e 2úi'' 1 ' ; : : : ; e 2úi'' n'
\Delta 2 T n \Psi ; ' 2 R;
where '' i = \Sigma1, i = 1; : : : ; n. Now define the automorphism / : T n ! T n by
/
\Gamma e 2úi'1 ; : : : ; e 2úi'n
\Delta =
\Gamma e 2úi'' 1 '1 ; : : : ; e 2úi'' n'n
\Delta :
It can be readily seen that / \Delta ` M = ` CP n , which together with Proposition 2.2.3
completes the proof.
Note that the equivariant diffeomorphism provided by Proposition 2.5.2 not nec­
essarily preserves the orientation of the quasitoric manifold M 2n .
The problem of equivariant and topological classification also admits a complete
solution in the case n = 2 (i.e. for quasitoric manifolds over polygons).
Example 2.5.3. The Hirzebruch surface is the 2­dimensional complex manifold
H p = CP (i p \Phi C ), where i p is the complex line bundle over CP 1 with first Chern
class p, C is the trivial complex line bundle and CP (\Delta) denotes the projectivisation

48 VICTOR M. BUCHSTABER AND TARAS E. PANOV
of the complex bundle. Hence, there is a bundle H p ! CP 1 with fibre CP 1 . The
surface H p is homeomorphic to S 2 \Theta S 2 for even p and to CP 2 # CP 2 for odd
p, where CP 2 denotes the space CP 2 with reversed orientation. The Hirzebruch
surfaces are non­singular projective toric varieties (see [40, p. 8]). The orbit space
for H p (regarded as a quasitoric manifold) is a combinatorial square; the corre­
sponding characteristic maps can be described using Example 2.2.8 (see also [33,
Example 1.19]).
Theorem 2.5.4 ([63, p. 553]). A quasitoric manifold of dimension 4 is equivari­
antly diffeomorphic to the equivariant connected sum of several copies of CP 2 and
Hirzebruch surfaces H p .
Corollary 2.5.5. A quasitoric manifold of dimension 4 is diffeomorphic to the
connected sum of several copies of CP 2 , CP 2 and S 2 \Theta S 2 .
The classification problem for quasitoric manifolds over a given simple polytope
can be considered as a generalisation of the corresponding problem for non­singular
toric varieties. In [62] to every toric variety over a simple 3­polytope P 3 was assigned
two integer weights on every edge of the dual simplicial complex KP . Using the
special ``monodromy conditions'' for weights, the complete classification of toric
varieties over simple 3­polytopes with Ÿ 8 facets was obtained in [62]. A similar
construction was used in [53] to obtain the classification of toric varieties over P n
with m = n + 2 facets (note that any such simple polytope is the product of two
simplices).
In [36] the construction of weights from [62] was generalised to the case of qua­
sitoric manifolds. This allowed to obtain a criterion [36, Theorem 3] for the existence
of a quasitoric manifold with given weight set and signs of vertices oe(v) (see Def­
inition 2.4.3; note that our signs of vertices correspond to the two­paint colouring
used in [36]). As an application, the complete classification of characteristic maps
for the cube I 3 is obtained in [36], along with a series of results on the classification
of quasitoric manifolds over the product of arbitrary number of simplices.
3. Moment­angle complexes
3.1. Moment­angle manifolds ZP defined by simple polytopes. For any
combinatorial simple polytope P n with m facets Davis and Januszkiewicz intro­
duced in [33] a space ZP with an action of the torus T m and the orbit space P n .
This space is universal for all quasitoric manifolds over P n in the sense that for every
quasitoric manifold ú : M 2n ! P n there is a principal T m\Gamman ­bundle ZP ! M 2n
whose composite map with ú is the orbit map for ZP . The space ZP and some its
generalisations turn to be very important and effective tool for studying different
combinatorial objects such as Stanley--Reisner rings, subspace arrangements, cubi­
cal complexes etc. In this section we reproduce the original definition of ZP and
adjust it in the way convenient for subsequent generalisations.
Let F = fF 1 ; : : : ; Fm g be the set of facets of P n . For any facet F i 2 F denote
by TF i the one­dimensional coordinate subgroup of T F ¸ = T m corresponding to F i .
Then assign to every face G the coordinate subtorus
TG =
M
F i oeG
TF i ae T F :

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 49
Note that dimTG = codimG. Recall that for any point q 2 P n we denoted by G(q)
the unique face containing q in the relative interior.
Definition 3.1.1. For any combinatorial simple polytope P n introduce the iden­
tification space
ZP = (T F \Theta P n )=¸;
where (t 1 ; q) ¸ (t 2 ; q) if and only if t 1 t \Gamma1
2 2 TG(q) .
The free action of T m on T F \Theta P n obviously descends to an action on ZP , with
quotient P n . Let ae : 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 ae \Gamma1 (v) with
maximal isotropy subgroup of dimension n.
Lemma 3.1.2. The space ZP is a smooth manifold of dimension m+ n.
In this paper we provide several different proofs of this lemma, each of which
arises from an equivalent definition of ZP . To give our first proof we need the
following simple topological fact.
Proposition 3.1.3. The torus T k admits a smooth embedding into R k+1 .
Proof. In order to construct a required embedding we provide a smooth function
g k+1 (x 1 ; : : : ; x k+1 ) such that the equation g k+1 = 0 defines a hypersurface diffeo­
morphic to T k . For k = 1 we take g 2 (x 1 ; x 2 ) = x 2
1 + (x 2 \Gamma 2) 2 \Gamma 1; then we proceed
by induction on k. Suppose we have a function g i (x 1 ; : : : ; x i ) such that fg i = 0g
defines a smooth embedding T i\Gamma1 ,! R i and x i ? 0 for any (x 1 ; : : : ; x i ) satisfying
g i = 0. Then set
g i+1 (x 1 ; : : : ; x i ; x i+1 ) := g i
\Gamma
x 1 ; : : : ; x i\Gamma1 ;
q
x 2
i + x 2
i+1
\Delta
:
The hypersurface fg i+1 = 0g ae R i+1 is easily seen to be diffeomorphic to T i .
Proof of Lemma 3.1.2. Construction 2.1.6 provides the atlas fU v g for P n as a man­
ifold with corners. The set U v is based on the vertex v and is diffeomorphic to R n
+ .
Then ae \Gamma1 (U v ) ¸ = T m\Gamman \Theta R 2n . We claim that T m\Gamman \Theta R 2n can be realised as an
open set in R m+n , thus providing a chart for ZP . To see this we embed T m\Gamman into
R m\Gamman+1 as a closed hypersurface H (Proposition 3.1.3). Since the normal bundle
is trivial, the small neighbourhood of H ae R m\Gamman+1 is homeomorphic to T m\Gamman \Theta R.
Taking the cartesian product with R 2n\Gamma1 we obtain an open set in R m+n homeo­
morphic to T m\Gamman \Theta R 2n .
The following statement follows easily from the definition of ZP .
Proposition 3.1.4. If P = P 1 \Theta P 2 for some simple polytopes P 1 , P 2 , then ZP =
ZP1 \Theta ZP2 . If G ae 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 2.2.2). 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 \Gamma n)­dimensional subtorus of T F .
Proposition 3.1.5. The subtorus H(`) acts freely on ZP , thus defining a principal
T m\Gamman ­bundle ZP !M 2n (`).

50 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Proof. It follows from (26) that H(`) meets every isotropy subgroup only at the
unit. This implies that the action of H(`) on ZP is free. By definitions of ZP and
M 2n (`), the projection – \Theta id : T F \Theta P n ! T n \Theta P n descends to the projection
(T F \Theta P n )=¸ \Gamma! (T n \Theta P n )=¸;
which displays ZP as a principal T m\Gamman ­bundle over M 2n (`).
To simplify notations we would write T m , C m etc. instead of T F , C F etc.
Define the unit poly­disk (D 2 ) m in the complex space as
(D 2 ) m =
\Phi (z 1 ; : : : ; z m ) 2 C m : jz i j Ÿ 1; i = 1; : : : ; m
\Psi :
Then (D 2 ) m is stable under the standard action of T m on C m , and the quotient is
the standard cube I m ae R m
+ .
Lemma 3.1.6. The cubical embedding i P : P n ! I m from Construction 1.5.5 is
covered by an equivariant embedding i e : ZP ! (D 2 ) m .
Proof. Recall that the cubical decomposition of P n consists of the cubes C n
v based
on the vertices v 2 P n . Note that C n
v is contained in the open set U v ae P n (see
Construction 2.1.6). The inclusion C n
v ae U v is covered by an equivariant inclusion
B v ae C m , where B v = ae \Gamma1 (C n
v ) is a closed subset homeomorphic to (D 2 ) m \Theta
T m\Gamman . Since ZP =
S
v2P n B v and B v is stable under the T m ­action, the resulting
embedding ZP ! (D 2 ) m is equivariant.
It follows from the proof that the manifold ZP is represented as a union of
fn\Gamma1 (P ) closed T m ­invariant charts B v . In section 3.3 we provide two different
ways to construct a cellular decomposition of each B v , thus describing ZP as a
cellular complex. For now, we mention that if v = F i 1
`` \Delta \Delta \Delta `` F i n , then
i e (B v ) = (D 2 ) n
i 1 ;::: ;i n
\Theta T m\Gamman
[m]nfi1 ;::: ;i ng
ae (D 2 ) m ;
or, more precisely,
i e (B v ) =
\Phi (z 1 ; : : : ; z m ) 2 (D 2 ) m : jz i j = 1 for i =
2 fi 1 ; : : : ; i n g
\Psi :
Recalling that the vertices of P n correspond to the maximal simplices of the poly­
topal sphere KP (boundary of the polar polytope P \Lambda ), we can write
i e (ZP ) =
[
I2KP
(D 2 ) I \Theta T [m]nI ae (D 2 ) m :
(37)
This can be regarded as an alternative definition of ZP . Introducing the polar
coordinates in (D 2 ) m we see that i e (B v ) is parametrised by n radial (or moment)
and m angle coordinates. That is why we refer to ZP as the moment­angle manifold
defined by P n .
Example 3.1.7. Let P n = \Delta n (the n­simplex). Then ZP is homeomorphic to the
(2n+1)­sphere S 2n+1 . The cubical complex C(\Delta n ) (see Construction 1.5.5) consists
of (n + 1) cubes C n
v . Each subset B v = ae \Gamma1 (C n
v ) is homeomorphic to (D 2 ) n \Theta S 1 .
In particular, for n = 1 we obtain the well­known representation of the 3­sphere S 3
as a union of two solid tori D 2 \Theta S 1 .
Another way to construct an equivariant embedding of ZP into C m can be de­
rived from Construction 1.1.4.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 51
Construction 3.1.8. Formula (6) defines the (affine) embedding AP : P n ,! R m
+ .
This embedding is covered by an equivariant embedding ZP ,! C m . A choice of
matrix W in Construction 1.1.4 defines a basis in the (m \Gamma n)­dimensional subspace
orthogonal to the n­plane that contain AP (P n ) (see (6)). The following statement
follows.
Corollary 3.1.9 (see also [26, x3]). The embedding ZP ,! C m has the trivial nor­
mal bundle.
3.2. General moment­angle complexes. In this section we extend the construc­
tion of ZP to the case of general simplicial complex K. The resulting space is not
a manifold for arbitrary K, but is so when K is a simplicial sphere.
Here we denote by ae the canonical projection (D 2 ) m ! I m , as well as any of
its restriction to a closed T m ­stable subset of (D 2 ) m . For each face C IaeJ of I m
(see (19)) define
(38) B IaeJ := ae \Gamma1 (C IaeJ )
= f(z 1 ; : : : ; z m ) 2 (D 2 ) m : z i = 0 for i 2 I; jz i j = 1 for i =
2 Jg:
It follows that if #I = i, #J = j then B IaeJ ¸ = (D 2 ) j \Gammai \Theta T m\Gammaj , where the disk
factors D 2 ae (D 2 ) j \Gammai are parametrised by J n I, while the circle factors S 1 ae T m\Gammaj
are parametrised by [m] n J .
Definition 3.2.1. Let C be a cubical subcomplex of I m . The moment­angle com­
plex ma(C) corresponding to C is the T m ­invariant decomposition of ae \Gamma1 (C) into
the ``moment­angle'' blocks B IaeJ (38) corresponding to the faces C IaeJ of C. Hence,
ma(C) is defined from the commutative diagram
ma(C) \Gamma\Gamma\Gamma\Gamma! (D 2 ) m
? ? y
? ? y ae
C \Gamma\Gamma\Gamma\Gamma! I m
:
The torus T m acts on ma(C) with orbit space C.
In section 1.5 two canonical cubical subcomplexes of I m , namely cub(K) (21)
and cc(K) (22), were associated to every simplicial complex K n\Gamma1 on m vertices.
We denote the corresponding moment­angle complexes by WK and ZK respectively.
Thus, we have
WK \Gamma\Gamma\Gamma\Gamma! (D 2 ) m
ae
? ? y
? ? y ae
cub(K) \Gamma\Gamma\Gamma\Gamma! I m
and
ZK \Gamma\Gamma\Gamma\Gamma! (D 2 ) m
ae
? ? y
? ? y ae
cc(K) \Gamma\Gamma\Gamma\Gamma! I m
;
(39)
where the horizontal arrows are embeddings, while the vertical ones are orbit maps
for T m ­actions. Note that dimZK = m + n and dimWK = m + n \Gamma 1.
Remark. Suppose that K = KP for some simple polytope P . Then it follows
from (37) that ZK is identified with ZP (or, more precisely, with i e (ZP )).
Lemma 3.2.2. If K is a simplicial (n \Gamma 1)­sphere, then ZK is an (m + n)­
dimensional (closed) manifold.

52 VICTOR M. BUCHSTABER AND TARAS E. PANOV
'
&
$
%
s s
0 1
I
T
D
(a)
'
&
$
%
s 1
T
D
(b)
Figure 7. Cellular decompositions of D 2 .
Proof. In this proof we identify the polyhedrons jKj and j cone(K)j with their
images cub(K) ae I m and cc(K) ae I m under the map j cone(K)j ! I m , see The­
orem 1.5.10. For each vertex fig 2 K denote by F i the union of (n \Gamma 1)­cubes of
cub(K) that contain fig. Alternatively, F i is j star bs(K) figj. These F 1 ; : : : ; Fm are
analogues of facets of a simple polytope. Moreover, if K = KP for some P , then F i
is the image of a facet of P under the map i P : C(P ) ! I m (Construction 1.5.5). As
in the case of simple polytopes, we define ``faces'' of cc(K) as non­empty intersec­
tions of ``facets'' F 1 ; : : : ; Fm . Then the ``vertices'' (i.e. non­empty intersections of n
``facets'') are the barycentres of (n \Gamma 1)­simplices of jKj. For every such barycentre b
denote by U b the open subset of cc(K) obtained by deleting all ``faces'' not contain­
ing b. Then U b is identified with R n
+ , while ae \Gamma1 (U b ) is homeomorphic to T m\Gamman \Theta R 2n .
This defines a structure of manifold with corners on the n­ball cc(K) = j cone(K)j,
with atlas fU b g. At the same time we see that ZK = ae \Gamma1 (cc(K)) is a manifold,
with atlas fae \Gamma1 (U b )g.
Problem 3.2.3. Characterise simplicial complexes K for which ZK is a manifold.
As we will see below (Theorem 4.2.1), for homological reasons, if ZK is a man­
ifold, then K is a Gorenstein* complex. So, the answer to the above problem is
somewhere between ``simplicial spheres'' and ``Gorenstein* complexes''.
3.3. Cellular structures on moment­angle complexes. Here we consider
two cellular decompositions of (D 2 ) m , which provide cellular decompositions for
moment­angle complexes. The first one has 5 m cells and define a cellular complex
structure (with 5 types of cells) for any moment­angle complex ma(C) ae (D 2 ) m .
The second cellular decomposition of (D 2 ) m has only 3 m cells, but is appropri­
ate only for defining a cellular structure (with 3 types of cells) on moment­angle
complexes ZK .
Let us consider the cellular decomposition of D 2 with one 2­cell D, two 1­cells
I, T , and two 0­cells 0, 1, see Figure 7 (a). It defines a cellular decomposition of
the poly­disk (D 2 ) m with 5 m cells. Each cell of this cellular complex is the product
of cells of 5 different types: D i , I i , 0 i , T i and 1 i , i = 1; : : : ; m. We encode the
cells of (D 2 ) m by ``words'' of type D I I J 0LTP 1Q , where I; J; L; P; Q are pairwise
disjoint subsets of [m] such that I [J [L[P [Q = [m]. Sometimes we would drop
the last factor 1Q , so in our notations D I I J 0LTP = D I I J 0LTP 1 [m]nI[J[L[P . The
closure of D I I J 0LTP 1Q is homeomorphic to the product of #I disks, #J segments,
and #P circles. The constructed cellular decomposition of (D 2 ) m allows to display
moment­angle complexes as certain cellular subcomplexes in (D 2 ) m .

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 53
Lemma 3.3.1. For any cubical subcomplex C of I m the corresponding moment­
angle complex ma(C) is a cellular subcomplex of (D 2 ) m .
Proof. Indeed, ma(C) is a union of ``moment­angle'' blocks B IaeJ (38), and each
B IaeJ is the closure of the cell D JnI I ; 0 I T [m]nJ 1 ; .
Now we concentrate on the moment­angle complex ZK corresponding to the
cubical complex cc(K) ae I m (see (39)). By definition, ZK is the union of moment­
angle blocks B IaeJ ae (D 2 ) m with J 2 K. Denote
B J := B ;aeJ =
\Phi (z 1 ; : : : ; z m ) 2 (D 2 ) m : jz j j = 1 for j =
2 J
\Psi :
(40)
Then B J = ae \Gamma1 (C J ) (recall that C J := C ;aeJ ) and B IaeJ ae B J for any I ae J . It
follows that
ZK =
[
J2K
B J
(41)
(compare this with the note after (22)).
Remark. If K = KP for a simple polytope P and #J = n, then B J is i e (B v ) for
v =
T
j2J F j . Hence, (41) reduces to (37) in this case.
Note that B J `` B J 0 = B J``J 0 . This allows to simplify the cellular decomposition
from Lemma 3.3.1 in the case ma(C) = ZK . To do this we replace the union of
cells 0, I, D (see Figure 7 (a)) by one 2­dimensional cell (which we also denote D).
The resulting cellular decomposition of D 2 with 3 cells is shown on Figure 7 (b).
It defines a cellular decomposition of (D 2 ) m with 3 m cells, each of which is the
product of 3 different types of cells: D i , T i and 1 i , i = 1; : : : ; m. We encode these
cells of (D 2 ) m as D I TP 1Q , where I; P; Q are pairwise disjoint subsets of [m] such
that I [P [Q = [m]. We use the notation D I TP := D I TP 1 [m]nI[P . The closure of
D I TP is the product of #I disks and #P circles.
Lemma 3.3.2. The moment­angle complex ZK is a cellular subcomplex of (D 2 ) m
with respect to the 3 m ­cell decomposition (see Figure 7 (b)). Each cell of ZK has
the form D I TP , I 2 K.
Proof. Since B J = B ;aeJ is the closure of the cell D J T [m]nJ 1 ; , the statement follows
from (41).
Remark. Note that for general C the moment­angle complex ma(C) is not a cellular
subcomplex for the 3 m ­cell decomposition of (D 2 ) m .
It follows from Construction 1.5.9 that the cubical complex cc(K) always con­
tains the vertex (1; : : : ; 1) 2 I m . Hence, the torus T m = ae \Gamma1 (1; : : : ; 1) is contained
in ZK .
Lemma 3.3.3. The inclusion T m = ae \Gamma1 (1; : : : ; 1) ,! ZK is a cellular map homo­
topical to the map to a point, i.e. the torus T m = ae \Gamma1 (1; : : : ; 1) is a contractible
cellular subcomplex of ZK .
Proof. To prove that T m = ae \Gamma1 (1; : : : ; 1) is a cellular subcomplex of ZK we just
mention that it is the closure of the m­cell D ; T [m] ae ZK . So, it remains to prove that
T m is contractible within ZK . To do this we show that the embedding T m ae (D 2 ) m
is homotopic to the map to the point (1; : : : ; 1) 2 T m ae (D 2 ) m . On the first step
we note that ZK contains the cell D 1 T 2;::: ;m , whose closure contains T m and is

54 VICTOR M. BUCHSTABER AND TARAS E. PANOV
homeomorphic to D 2 \Theta T m\Gamma1 . Hence, our T m can be contracted to 1 \Theta T m\Gamma1
within ZK . On the second step we note that ZK contains the cell D 2 T 3;::: ;m , whose
closure contains 1 \Theta T m\Gamma1 and is homeomorphic to D 2 \Theta T m\Gamma2 . Hence, 1 \Theta T m\Gamma1
can be contracted to 1 \Theta 1 \Theta T m\Gamma2 within ZK , and so on. On the k­th step we note
that ZK contains the cell D k T k+1;::: ;m , whose closure contains 1 \Theta \Delta \Delta \Delta \Theta 1 \Theta T m\Gammak+1
and is homeomorphic to D 2 \Theta T m\Gammak . Hence, 1 \Theta \Delta \Delta \Delta \Theta 1 \Theta T m\Gammak+1 can be contracted
to 1 \Theta \Delta \Delta \Delta \Theta 1 \Theta T m\Gammak within ZK . We end up at the point 1 \Theta \Delta \Delta \Delta \Theta 1 to which the
whole torus T m can be contracted.
Corollary 3.3.4. For any simplicial complex K the moment­angle complex ZK is
simply connected.
Proof. Indeed, the 1­skeleton of our cellular decomposition of ZK is contained in
the torus T m = ae \Gamma1 (1; : : : ; 1).
3.4. Borel construction and Stanley--Reisner space. Let ET m be the con­
tractible space of the universal principal T m ­bundle over the classifying space BT m .
It is well known that BT m is (homotopy equivalent to) the product of m copies
of infinite­dimensional projective space CP 1 . The cellular decomposition of CP 1
with one cell in every even dimension determines the canonical cellular decomposi­
tion of BT m . It follows that the cohomology of BT m (with coefficients in k) is the
polynomial ring k[v 1 ; : : : ; vm ], deg v i = 2.
Definition 3.4.1. Let X be a space with an action of the torus T m . The Borel
construction (alternatively, homotopy quotient or associated bundle) is the identi­
fication space
ET m \Theta T m X := ET m \Theta X=¸;
where (e; x) ¸ (eg; g \Gamma1 x) for any e 2 ET m , x 2 X, g 2 T m .
The projection (e; x) ! e displays ET m \Theta T m X as the total space of a bundle
ET m \Theta T m X ! BT m with fibre X and structure group T m . At the same time,
there is a principal T m ­bundle ET m \Theta X ! ET m \Theta T m X.
In the sequel we denote the Borel construction ET m \Theta T m X corresponding to
a T m ­space X by BT X. In particular, for any simplicial complex K on m vertices
there are defined the Borel construction B T ZK and the bundle p : BT ZK ! BT m
with fibre ZK .
For each i = 1; : : : ; m denote by BT i the i­th coordinate subspace of BT m =
(C P 1 ) m . For any subset I ae [m], we denote by BT I the product of BT i with i 2 I.
Obviously, BT I is a cellular subcomplex of BT m , and BT I ¸ = BT k provided that
#I = k.
Definition 3.4.2. Let K be a simplicial complex. We refer to the cellular subcom­
plex [
I2K
BT I ae BT m
as the Stanley--Reisner space, and denote it SR(K).
The name refers to the following statement, which is an immediate corollary of the
definition of Stanley--Reisner ring k(K) (Definition 1.3.1).
Proposition 3.4.3. The cellular cochain algebra C \Lambda (SR(K)) and the cohomology
algebra H \Lambda (SR(K)) are isomorphic to the face ring k(K). The cellular inclusion

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 55
i : SR(K) ,! BT m induces the quotient epimorphism i \Lambda : k[v 1 ; : : : ; vm ] ! k(K) =
k[v 1 ; : : : ; v m ]=I K in the cohomology.
Theorem 3.4.4. The fibration p : B T ZK ! BT m is homotopy equivalent to the
cellular inclusion i : SR(K) ,! BT m . More precisely, there is a deformation re­
traction BT ZK ! SR(K) such that the diagram
BT ZK p
\Gamma\Gamma\Gamma\Gamma! BT m
? ? y
fl fl fl
SR(K) i
\Gamma\Gamma\Gamma\Gamma! BT m
is commutative.
Proof. Consider the decomposition (41). Since each B J ae ZK is T m ­stable, the
Borel construction BT ZK = ET m \Theta T m ZK is patched from the Borel constructions
ET m \Theta T m B J , J 2 K. Suppose #J = j; then B J ¸ = (D 2 ) j \Theta T m\Gammaj (see (40)). By
definition of Borel construction, ET m \Theta T m B J ¸ = (ET j \Theta T j (D 2 ) j ) \Theta ET m\Gammaj . The
space ET j \Theta T j (D 2 ) j is the total space of a (D 2 ) j ­bundle over BT j . It follows that
there is a deformation retraction ET m \Theta T m B J ! BT J , which defines a homotopy
equivalence between the restriction of p : B T ZK ! BT m to ET m \Theta T m B J and
the cellular inclusion BT J ,! BT m . These homotopy equivalences corresponding
to different simplices J 2 K fit together to yield a required homotopy equivalence
between p : BT ZK ! BT m and i : SR(K) ,! BT m .
Corollary 3.4.5. The moment­angle complex ZK is a homotopy fibre of the cellu­
lar inclusion i : SR(K) ,! BT m .
Corollary 3.4.6. The cohomology algebra H \Lambda (B T ZK ) is isomorphic to the face
ring k(K). The projection p : B T ZK ! BT m induces the quotient epimorphism
p \Lambda : k[v 1 ; : : : ; vm ] ! k(K) = k[v 1 ; : : : ; vm ]=I K in the cohomology.
Remark. The above statement was proved in [33, Theorem 4.8] in the polytopal
case (ZK = ZP ) by other methods.
The following information about the homotopy groups of ZK can be retrieved
from the above constructions.
Theorem 3.4.7. (a) The complex ZK is 2­connected (i.e. ú 1 (ZK ) = ú 2 (ZK ) = 0),
and ú i (ZK ) = ú i (BT ZK ) = ú i (SR(K)) for i – 3.
(b) If K = KP and P is q­neighbourly, then ú i (ZK ) = 0 for i ! 2q + 1, and
ú 2q+1 (ZP ) is a free Abelian group whose generators correspond to the (q+1)­element
missing faces of KP (or equivalently, to degree­(2q + 2) generators of the ideal IP ,
see Definition 1.1.20).
Proof. Note that BT m = K(Z m ; 2) and the 3­skeleton of SR(K) coincides with
that of BT m . If P is q­neighbourly, then it follows from Definition 3.4.2 that the
(2q + 1)­skeleton of SR(KP ) coincides with that of BT m . Now, both statements
follow easily from the exact homotopy sequence of the map i : SR(K) ! BT m
with homotopy fibre ZK (Corollary 3.4.5).
Remark. Say that a simplicial complex K on the set [m] is k­neighbourly if any k­
element subset of [m] is a simplex of K. (This definition is an obvious extension of
the notion of k­neighbourly simplicial polytope to arbitrary simplicial complexes).

56 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Then the second part of Theorem 3.4.7 holds for arbitrary q­neighbourly simplicial
complex.
Suppose now that K = KP for some simple n­polytope P and M 2n is a qua­
sitoric manifold over P with characteristic function `. Then we have the subgroup
H(`) ae T m acting freely on ZP and the principal T m\Gamman ­bundle ZP !M 2n (Propo­
sition 3.1.5).
Proposition 3.4.8. The Borel construction ET n \Theta T n M 2n is homotopy equivalent
to BT ZP .
Proof. Since H(`) acts freely on ZP , we have
BT ZP = ET m \Theta T m ZP
= EH(`) \Theta
i
E
\Gamma T m =H(`)
\Delta \Theta T m =H(`) ZP =H(`)
j
' ET n \Theta T n M 2n :
Theorem 3.4.9 ([33, Theorem 4.12]). The Leray--Serre spectral sequence of the
bundle
ET n \Theta T n M 2n ! BT n
(42)
with fibre M 2n collapses at the E 2 term, i.e. E p;q
2 = E p;q
1 .
Proof. The differentials of the spectral sequence are all trivial by dimensional rea­
sons, since both BT n and M 2n have cells only in even dimensions (see Proposi­
tion 2.2.5).
Corollary 3.4.10. The projection (42) induces a monomorphism k[t 1 ; : : : ; t n ] !
k(P ) in the cohomology. The inclusion of the fibre M 2n ,! ET n \Theta T n M 2n induces
an epimorphism k(P ) ! H \Lambda (M 2n ).
Now we are ready to prove the statements from section 2.2.
Proof of Lemma 2.2.6 and Theorem 2.2.7. The monomorphism
H \Lambda (BT n ) = k[t 1 ; : : : ; t n ] ! k(P ) = H \Lambda (ET n \Theta T n M 2n )
takes t i to ` i , i = 1; : : : ; n. Since k(P ) is a free k[t 1 ; : : : ; t n]­module (note that this
follows from Theorem 3.4.9, so we do not need to use Theorem 1.3.8), ` 1 ; : : : ; ` n
is a regular sequence. It follows that the kernel of k(P ) ! H \Lambda (M 2n ) is exactly
J ` = (` 1 ; : : : ; ` n ).
3.5. Generalisations, analogues and additional comments. Many impor­
tant constructions from our survey (namely, the cubical complex cc(K), the
moment­angle complex ZK , the Borel construction B T ZK , the Stanley--Reisner
space SR(K), and also the complement U (K) of a coordinate subspace arrange­
ment from section 5.2) admit the unifying combinatorial interpretation in terms
of the following construction, which was proposed by N. Strickland (in private
communications).
Construction 3.5.1. Let X be a space, and W a subspace of X. Let K be a
simplicial complex on the vertex set [m]. Define the following subset of the product
of m copies of X:
K ffl (X; W ) =
[
I2K
iY
i2I
X \Theta
Y
i =
2I
W
j
:

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 57
Example 3.5.2. 1. cc(K) = K ffl (I 1 ; 1) (see (22)).
2. ZK = K ffl (D 2 ; S 1 ) (see (41)).
3. SR(K) = K ffl (C P 1 ; \Lambda) (see Definition 3.4.2).
4. B T ZK = K ffl (ES 1 \Theta S 1 D 2 ; ES 1 \Theta S 1 S 1 ) (see the proof of Theorem 3.4.4).
Note that Construction 3.5.1 is obviously extended to the set of m pairs
(X 1 ; W 1 ); : : : ; (Xm ; Wm ); another generalisation is obtained by replacing the carte­
sian product by the fibred product.
Using construction 3.5.1, the part of Theorem 1.5.10 (dealing with the com­
plex cc(K)) and Theorem 5.2.5 were obtained independently by N. Strickland.
Almost all constructions of our paper incorporate some action of the torus, i.e.
the product of circles S 1 . These constructions admit natural Z=2­analogues. To see
this we replace the torus T m by its ``real analogue'', that is, the group (Z=2) m .
Then the standard cube I m = [0; 1] m is the orbit space for the action of (Z=2) m on
the bigger cube [\Gamma1; 1] m , which can be regarded as a ``real analogue'' of the poly­
disk (D 2 ) m ae C m . Now, given a cubical subcomplex C ae I m , we can construct a
(Z=2) m ­symmetrical cubical complex embedded into [\Gamma1; 1] m just in the same way
as we did it in Definition 3.2.1. In particular, for any simplicial complex K on the
vertex set [m] we can introduce the cubical complexes RZK and RWK , the ``real
analogues'' of the moment­angle complexes ZK and WK (39). In the notations of
Construction 3.5.1 we have
RZK = K ffl
\Gamma [\Gamma1; 1]; f\Gamma1; 1g
\Delta :
If K is a simplicial (n \Gamma 1)­sphere, then RZK is an n­dimensional manifold (the
proof is identical to that of Lemma 3.2.2). Thus, for any simplicial sphere K n\Gamma1
with m vertices we get a (Z=2) m ­symmetric n­manifold with a (Z=2) m ­invariant
cubical subdivision. This class of cubical manifolds may be useful for the cubical
analogue for the combinatorial theory of f­vectors of simplicial complexes (see
also [82]). Finally, the real analogue RZP of the manifold ZP (corresponding to the
case of polytopal simplicial sphere K n\Gamma1 ) is known as the universal Abelian cover
of the polytope P n regarded as an orbifold (or manifold with corners), see [43, x4.5].
In [47] manifolds RZP and ZP are interpreted as the configuration spaces of hinge
mechanisms in R 2 and R 3 .
Passage from T n to (Z=2) n in Definition 2.2.1 leads to real analogues of quasitoric
manifolds, which were introduced in [33] under the name small covers. Thus, every
small cover of a simple polytope P n is a manifold M n with an action of (Z=2) n
and quotient P n . The name refers to the fact that any branched cover of P n by a
smooth manifold have at least 2 n sheets. Small covers were studied in [33] along
with quasitoric manifolds, and many results on quasitoric manifolds cited from [33]
in section 2.2 have analogues in the small cover case. On the other hand, every
small cover is the quotient of the universal cover RZP by a certain free action of
the group (Z=2) m\Gamman .
An important class of small covers over 3­dimensional simple polytopes P 3 was
considered in [48]. It can be shown that a simple polytope P 3 admits a 3­paint
colouring of its facets such that any two adjacent facets have different colour if and
only if every facet have even number of edges. Every such colouring % defines the
quasitoric manifold M 6 (%) and the small cover M 3 (%). It was shown in [48] that
every manifold M 3 (%) admits an equivariant embedding into R 4 = R 3 \Theta R with
the standard action of (Z=2) 3 on R 3 and the trivial action on R. It was also shown

58 VICTOR M. BUCHSTABER AND TARAS E. PANOV
there that all manifolds M 3 (%) can be obtained from the set of 3­dimensional tori by
applying the operations of equivariant connected sum and equivariant Dehn twist .
The quaternionic analogue of moment­angle complexes can be constructed by
replacing T n by the quaternionic torus Sp(1) n ¸ = (S 3 ) n . Developing the quaternionic
analogues of toric and quasitoric manifolds is a problem of particular interest. This
is acknowledged, in particular, by the important results of [16].
At the end we give one application of the above described constructions to the
case of general group G.
Example 3.5.3 (classifying space for group G). Let K be a simplicial complex on
the vertex set [m]. Set ZK (G) := K ffl (cone(G); G) (see Construction 3.5.1), where
cone(G) is the cone over G. By the construction, the group G m acts on ZK (G)
with quotient cone(K). The diagonal embedding G ,! G m defines a free action of
G on ZK (G).
Suppose now that K 0 ae K 1 ae \Delta \Delta \Delta ae K i ae \Delta \Delta \Delta is a sequence of embedded sim­
plicial complexes such that K i is q i ­neighbourly and q i ! 1 as i ! 1. Such a
sequence can be constructed, for instance, by taking K i+1 := K i \Lambda K (see Con­
struction 1.2.5), where K 0 and K are arbitrary simplicial complexes. The group
G acts freely on the space lim \Gamma! ZK i (G), and the corresponding quotient provides a
realisation of the classifying space BG. Thus, we have the following filtration in the
universal fibration EG ! BG:
ZK0 (G) ,! ZK1 (G) ,! \Delta \Delta \Delta ,! ZK i (G) ,! \Delta \Delta \Delta
# # #
ZK0 (G)=G ,! ZK1 (G)=G ,! \Delta \Delta \Delta ,! ZK i (G)=G ,! \Delta \Delta \Delta :
The well­known Milnor filtration in the universal fibration of the group G corre­
sponds to the case K i = \Delta i .
4. Cohomology of moment­angle complexes and combinatorics of
simplicial manifolds
4.1. Eilenberg--Moore spectral sequence. In their 1966 paper [39] Eilenberg
and Moore constructed a spectral sequence of great importance for algebraic topol­
ogy. This spectral sequence can be considered as an extension of Adams' approach
to calculating the cohomology of loop spaces [1]. In 1960­70s the Eilenberg--Moore
spectral sequence allowed to obtain many important results on the cohomology of
loop spaces and homogeneous spaces for Lie group actions. In our paper we describe
new applications of this spectral sequence to combinatorial problems. This section
contains the necessary information about the spectral sequence; we follow Smith's
paper [76] in this description.
To be precise, there are two Eilenberg--Moore spectral sequences, the algebraic
and the topological.
Theorem 4.1.1 (Eilenberg--Moore [76, Theorem 1.2]). Let A be a differential
graded k­algebra, and M , N differential graded A­modules. Then there exists a
spectral sequence fE r ; d r g that converges to TorA (M; N ) and has the E 2 ­term
E \Gammai;j
2 = Tor \Gammai;j
H[A]
\Gamma H[M ]; H[N ]
\Delta ; i; j – 0;
where H[\Delta] denotes the cohomology algebra (module).

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 59
The spectral sequence of Theorem 4.1.1 lives in the second quadrant and
the differential d r adds (r; 1 \Gamma r) to bidegree. This spectral sequence is called
the (algebraic) Eilenberg--Moore spectral sequence. It gives a decreasing filtration
fF \Gammap TorA (M; N )g on TorA (M; N ) with the property that
E \Gammap;n+p
1 = F \Gammap
` X
\Gammai+j=n
Tor \Gammai;j
A (M; N )
'OE
F \Gammap+1
` X
\Gammai+j=n
Tor \Gammai;j
A (M;N )
'
:
Topological applications of Theorem 4.1.1 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
E \Gamma\Gamma\Gamma\Gamma! E 0
? ? y
? ? y
B \Gamma\Gamma\Gamma\Gamma! B 0 ;
(43)
where E 0 ! B 0 is a Serre fibre bundle with fibre F over the simply connected
base B 0 , and E ! B is the pullback along a continuous map B ! B 0 . For any
space X, let C \Lambda (X) denote either the singular cochain algebra of X or (in the case
when X is a cellular complex) the cellular cochain algebra of X. Obviously, C \Lambda (E 0 )
and C \Lambda (B) are C \Lambda (B 0 )­modules. Under these assumptions the following statement
holds.
Lemma 4.1.2 ([76, Proposition 3.4]). Tor C \Lambda (B0 ) (C \Lambda (E 0 ); C \Lambda (B)) is an algebra in
a natural way, and the is a canonical isomorphism of algebras
Tor C \Lambda (B0 )
\Gamma
C \Lambda (E 0 ); C \Lambda (B)
\Delta ! H \Lambda (E):
Applying Theorem 4.1.1 in the case A = C \Lambda (B 0 ), M = C \Lambda (E 0 ), N = C \Lambda (B) and
taking into account Lemma 4.1.2 we obtain
Theorem 4.1.3 (Eilenberg--Moore). There exists a spectral sequence of commuta­
tive algebras fE r ; d r g with
(a) E r ) H \Lambda (E);
(b) E \Gammai;j
2 = Tor \Gammai;j
H \Lambda (B0 ) (H \Lambda (E 0 ); H \Lambda (B)).
The spectral sequence of Theorem 4.1.3 is called the (topological) Eilenberg--
Moore spectral sequence. In the very important particular case when B is a point
(see (43)) we get
Corollary 4.1.4. Let E ! B be a fibration over the simply connected space B with
fibre F . Then there exists a spectral sequence of commutative algebras fE r ; d r g with
(a) E r ) H \Lambda (F ),
(b) E 2 = Tor H \Lambda (B) (H \Lambda (E); k).
We refer to the spectral sequence of Corollary 4.1.3 as the Eilenberg--Moore spec­
tral sequence of fibration E ! B.
Example 4.1.5. Let M 2n be a quasitoric manifoldover P n . Consider the Eilenberg--
Moore spectral sequence of the bundle ET n \Theta T n M 2n ! BT n with fibre M 2n . By
Proposition 3.4.8, H \Lambda (ET n \Theta T n M 2n ) = H \Lambda (BT ZP ) ¸ = k(P n ). The monomorphism
k[t 1 ; : : : ; t n ] = H \Lambda (BT n ) ! H \Lambda (ET n \Theta T n M 2n ) = k(P n )

60 VICTOR M. BUCHSTABER AND TARAS E. PANOV
takes t i to ` i , i = 1; : : : ; n, see (27). The E 2 term of the Eilenberg--Moore spectral
sequence is
E \Lambda;\Lambda
2 = Tor \Lambda;\Lambda
H \Lambda (BT n )
\Gamma H \Lambda (ET n \Theta T n M 2n ); k
\Delta = Tor \Lambda;\Lambda
k[t1 ;::: ;t n ]
\Gamma k(P n ); k
\Delta :
Since k(P n ) is a free k[t 1 ; : : : ; t n]­module, we have
Tor \Lambda;\Lambda
k[t1 ;::: ;t n ]
\Gamma k(P n ); k
\Delta = Tor 0;\Lambda
k[t1 ;::: ;t n ]
\Gamma k(P n ); k
\Delta
= k(P n
)\Omega k[t1 ;::: ;t n ] k = k(P n )=(` 1 ; : : : ; ` n ):
Therefore, E 0;\Lambda
2 = k(P n )=J ` and E \Gammap;\Lambda
2 = 0 for p ? 0. It follows that the Eilenberg--
Moore spectral sequence collapses at the E 2 term and H \Lambda (M 2n ) = k(P n )=J ` , in
accordance with Theorem 2.2.7.
4.2. Cohomology of moment­angle complex ZK : the case of general K.
Here we apply the Eilenberg--Moore spectral sequence to calculating the cohomology
algebra of the moment­angle complex ZK . This describes H \Lambda (ZK ) as a bigraded
algebra. The corresponding bigraded Betti numbers are important combinatorial
invariants of K.
Theorem 4.2.1. The following isomorphism of graded algebras holds:
H \Lambda (ZK ) ¸ = Tor k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta
:
In particular,
H p (ZK ) ¸ =
X
\Gammai+2j=p
Tor \Gammai;2j
k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta :
Proof. Let us consider the Eilenberg--Moore spectral sequence of the commutative
square
E \Gamma\Gamma\Gamma\Gamma! ET m
? ? y
? ? y
SR(K) i
\Gamma\Gamma\Gamma\Gamma! BT m ;
(44)
where the left vertical arrow is the pullback along i. Corollary 3.4.5 shows that E
is homotopy equivalent to ZK .
By Proposition 3.4.3, the map i : SR(K) ,! BT m induces the quotient epimor­
phism i \Lambda : C \Lambda (BT m ) = k[v 1 ; : : : ; vm ] ! k(K) = C \Lambda (SR(K)), where C \Lambda (\Delta) denotes
the cellular cochain algebra. Since ET m is contractible, there is a chain equivalence
C \Lambda (ET m ) ' k. Therefore, there is an isomorphism
Tor C \Lambda (BT m )
\Gamma C \Lambda (SR(K)); C \Lambda (ET m )
\Delta ¸ = Tor k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta :
(45)
The Eilenberg--Moore spectral sequence of commutative square (44) has
E 2 = Tor H \Lambda (BT m )
\Gamma H \Lambda (SR(K)); H \Lambda (ET m )
\Delta
and converges to Tor C \Lambda (BT m ) (C \Lambda (SR(K)); C \Lambda (ET m )) (Theorem 4.1.1). Since
Tor H \Lambda (BT m )
\Gamma H \Lambda (SR(K)); H \Lambda (ET m )
\Delta = Tor k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta ;
it follows from (45) that the spectral sequence collapses at the E 2 term, that is, E 2 =
E1 . Lemma 4.1.2 shows that the module Tor C \Lambda (BT m ) (C \Lambda (SR(K)); C \Lambda (ET m )) is an
algebra isomorphic to H \Lambda (ZK ), which concludes the proof.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 61
Theorem 4.2.1 describes the cohomology of ZK as a bigraded algebra and shows
that the corresponding bigraded Betti numbers b \Gammai;2j (ZK ) coincide with that
of k(K), see (17). Using the Koszul resolution for k and Lemma 1.4.6, we get
Theorem 4.2.2. The following isomorphism of bigraded algebras holds:
H \Lambda;\Lambda (ZK ) ¸ = H
\Theta \Lambda[u 1 ; : : : ; um
]\Omega k(K); d
\Lambda ;
where the bigraded structure and the differential in the right hand side are defined
by (16).
In the sequel we denote square­free monomials
u i 1 : : : u i p v j1 : : : v jq 2 \Theta \Lambda[u 1 ; : : : ; um
]\Omega k(K); d
\Lambda
by u I v J , where I = fi 1 ; : : : ; i p g, J = fj 1 ; : : : ; j q g. Note that bideg u I v J =
(\Gammap; 2(p + q)).
Remark. Since the differential d in (16) does not change the second degree, the dif­
ferential bigraded algebra [\Lambda[u 1 ; : : : ; um
]\Omega k(K); d] splits into the sum of differential
algebras consisting of elements of fixed second degree.
Corollary 4.2.3. The Leray--Serre spectral sequence of the principal T m ­bundle
ET m \Theta ZK ! BT ZK collapses at the E 3 term.
Proof. The spectral sequence under consideration converges to H \Lambda (ET m \Theta ZK ) =
H \Lambda (ZK ) and has
E 2 = H \Lambda (B T ZK
)\Omega H \Lambda (T m ) = \Lambda[u 1 ; : : : ; um
]\Omega k(K):
The differential in the E 2 term acts as in (16). Hence,
E 3 = H[E 2 ; d] = H
\Theta \Lambda[u 1 ; : : : ; um
]\Omega k(K)
\Lambda = H \Lambda (ZK )
(the last identity follows from Theorem 4.2.2).
Construction 4.2.4. Let A \Gammaq (K) ae \Lambda[u 1 ; : : : ; um
]\Omega k(K) be the subspace gen­
erated by monomials u I and u I v J such that J is a simplex of K, #I = q and
I `` J = ;. Define
A \Lambda (K) =
m
M
q=0
A \Gammaq (K):
Since d(u i ) = v i , we have d
\Gamma A \Gammaq (K)
\Delta ae A \Gammaq+1 (K). Therefore, A \Lambda (K) is a cochain
subcomplex in [\Lambda[u 1 ; : : : ; um
]\Omega k(K); d]. Moreover, A \Lambda (K) inherits the bigraded
module structure from \Lambda[u 1 ; : : : ; um
]\Omega k(K), with differential d adding (1; 0) to
bidegree. Hence, we have the additive inclusion (i.e. the monomorphism of bi­
graded modules) i a : A \Lambda (K) ,! \Lambda[u 1 ; : : : ; um
]\Omega k(K). Finally, A \Lambda (K) is an al­
gebra in the obvious way, but is not a subalgebra of \Lambda[u 1 ; : : : ; um
]\Omega k(K). (In­
deed, for instance, v 2
1 = 0 in A \Lambda (K) but not in \Lambda[u 1 ; : : : ; um
]\Omega k(K).) Neverthe­
less, we have the multiplicative projection (an epimorphism of bigraded algebras)
j m : \Lambda[u 1 ; : : : ; um
]\Omega k(K) ! A \Lambda (K). The additive inclusion i a and the multiplica­
tive projection j m obviously satisfy j m \Delta i a = id.
Lemma 4.2.5. Cochain complexes [\Lambda[u 1 ; : : : ; um
]\Omega k(K); d] and [A \Lambda (K); d] have
the same cohomology. This implies the following isomorphism of bigraded k­
modules:
H[A \Lambda (K); d] ¸ = Tor k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta
:

62 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Proof. A routine check shows that the cochain homotopy operator s for the Koszul
resolution (see the proof of Proposition VII.2.1 in [56]) establishes a cochain homo­
topy equivalence between the maps id and i a \Delta j m of the algebra [\Lambda[u 1 ; : : : ; um
]\Omega k(K); d] to itself. That is,
ds + sd = id \Gamma i a \Delta j m :
We just illustrate the above identity on few simple examples.
1) s(u 1 v 2 ) = u 1 u 2 ; ds(u 1 v 2 ) = u 2 v 1 \Gamma u 1 v 2 ; sd(u 1 v 2 ) = u 1 v 2 \Gamma u 2 v 1 ;
hence, (ds + sd)(u 1 v 2 ) = 0 = (id \Gamma i a \Delta j m )(u 1 v 2 );
2) s(u 1 v 1 ) = u 2
1 = 0; ds(u 1 v 1 ) = 0; d(u 1 v 1 ) = v 2
1 ; sd(u 1 v 1 ) = u 1 v 1 ;
hence, (ds + sd)(u 1 v 1 ) = u 1 v 1 = (id \Gamma i a \Delta j m )(u 1 v 1 );
3) s(v 2
1 ) = u 1 v 1 ; ds(v 2
1 ) = v 2
1 ; d(v 2
1 ) = 0;
hence, (ds + sd)(v 2
1 ) = v 2
1 = (id \Gamma i a \Delta j m )(v 2
1 ):
Now we recall our cellular decomposition of ZK , see Lemma 3.3.2. The cells are
D I T J with J ae [m], I 2 K, and I `` J = ;. Let C \Lambda (ZK ) and C \Lambda (ZK ) denote the
corresponding cellular chain and cochain complexes respectively. Both complexes
C \Lambda (ZK ) and A \Lambda (K) have the same cohomology H \Lambda (ZK ). The complex C \Lambda (ZK ) has
the basis consisting of cochains (D I T J ) \Lambda . As an algebra, C \Lambda (ZK ) is generated by
the cochains T \Lambda
i , D \Lambda
i of dimension 1 and 2 respectively dual to the cells T i and D i ,
i; j = 1; : : : ; m. At the same time, A \Lambda (K) is multiplicatively generated by u i , v i ,
i; j = 1; : : : ; m.
Theorem 4.2.6. The correspondence v I u J 7! (D I T J ) \Lambda establishes a canonical iso­
morphism of differential graded algebras A \Lambda (K) and C \Lambda (ZK ).
Proof. It follows directly from the definitions of A \Lambda (K) and C \Lambda (ZK ) that the map
is an isomorphism of graded algebras. Thus, it remains to prove that it commutes
with differentials. Let d, d c and @ c denote the differentials in A \Lambda (K), C \Lambda (ZK ) and
C \Lambda (ZK ) respectively. Since d(v i ) = 0, d(u i ) = v i , we need to show that d c (D \Lambda
i ) = 0,
d c (T \Lambda
i ) = D \Lambda
i . We have @ c (D i ) = T i , @ c (T i ) = 0. Any 2­cell of ZK is either D j or
T jk , k 6= j. Then
(d c T \Lambda
i ; D j ) = (T \Lambda
i ; @ c D j ) = (T \Lambda
i ; T j ) = ffi ij ; (d c T \Lambda
i ; T jk ) = (T \Lambda
i ; @ c T jk ) = 0;
where ffi ij = 1 if i = j and ffi ij = 0 otherwise. Hence, d c (T \Lambda
i ) = D \Lambda
i . Further, any
3­cell of ZK is either D j T k or T j1 j2 j3 . Then
(d c D \Lambda
i ; D j T k ) = (D \Lambda
i ; @ c (D j T k )) = (D \Lambda
i ; T jk ) = 0;
(d c D \Lambda
i ; T j1 j2 j3 ) = (D \Lambda
i ; @ c T j1 j2 j3 ) = 0:
Hence, d c (D \Lambda
i ) = 0.
Theorem 4.2.6 provides a topological interpretation for the differential algebra
[A \Lambda (K); d]. In the sequel we do not distinguish the cochain complexes A \Lambda (K) and
C \Lambda (ZK ), and identify u i with T \Lambda
i , v i with D \Lambda
i .
Now we recall that the algebra [A \Lambda (K); d] is bigraded. The isomorphism of Theo­
rem 4.2.6 provides a bigraded structure for the cellular chain complex C \Lambda (ZK ); @ c ],

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 63
with
bideg(D i ) = (0; 2); bideg(T i ) = (\Gamma1; 2); bideg(1 i ) = (0; 0):
(46)
The differential @ c adds (\Gamma1; 0) to bidegree, and the cellular homology of ZK also
acquires a bigraded structure.
Let us assume now that the ground field k is of zero characteristic (e.g. k = Q,
the field of rational numbers). Define the bigraded Betti numbers
b \Gammaq;2p (ZK ) = dimH \Gammaq;2p [C \Lambda (ZK ); @ c ]; q; p = 0; : : : ; m:
(47)
Theorem 4.2.6 and Lemma 4.2.5 show that
b \Gammaq;2p (ZK ) = dimTor \Gammaq;2p
k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta = fi \Gammaq;2p \Gamma k(K)
\Delta
(48)
(see (17)). Alternatively, b \Gammaq;2p (ZK ) equals the dimension of (\Gammaq; 2p)­th bigraded
component of the cohomology algebra H [\Lambda[u 1 ; : : : ; um
]\Omega k(K); d]). For the ordinary
Betti numbers b k (ZK ) holds
b k (ZK ) =
X
\Gammaq+2p=k
b \Gammaq;2p (ZK ); k = 0; : : : ; m+ n:
(49)
Below we describe some basic properties of bigraded Betti numbers (47).
Lemma 4.2.7. Let K n\Gamma1 be a simplicial complex with m = f 0 vertices and f 1
edges, and let ZK be the corresponding moment­angle complex, dimZK = m + n.
Then
(a) b 0;0 (ZK ) = b 0 (ZK ) = 1, b 0;2p (ZK ) = 0 for p ? 0;
(b) b \Gammaq;2p = 0 for p ? m or q ? p;
(c) b 1 (ZK ) = b 2 (ZK ) = 0;
(d) b 3 (ZK ) = b \Gamma1;4 (ZK ) =
\Gamma f0
2
\Delta \Gamma f 1 ;
(e) b \Gammaq;2p (ZK ) = 0 for q – p ? 0 or p \Gamma q ? n;
(f) b m+n (ZK ) = b \Gamma(m\Gamman);2m (ZK ).
Proof. We make calculations with the cochain complex A \Lambda (K) ae \Lambda[u 1 ; : : : ; um
]\Omega k(K). The module A \Lambda (K) has the basis consisting of monomials u J v I with I 2 K
and I `` J = ;. Since bideg v i = (0; 2), bideg u j = (\Gamma1; 2), the bigraded component
A \Gammaq;2p (K) is generated by monomials u J v I with #I = p \Gamma q and #J = q. In
particular, A \Gammaq;2p (K) = 0 if p ? m or q ? p, whence the assertion (b) follows.
To prove (a) we observe that A 0;0 (K) is generated by 1, while any v I 2 A 0;2p (K),
p ? 0, is a coboundary, whence H 0;2p (ZK ) = 0, p ? 0.
Now we prove the assertion (e). Any u J v I 2 A \Gammaq;2p (K) has I 2 K, while any
simplex of K is at most (n\Gamma1)­dimensional. It follows that A \Gammaq;2p (K) = 0 for p\Gammaq ?
n. By (b), b \Gammaq;2p (ZK ) = 0 for q ? p, so it remains to prove that b \Gammaq;2q (ZK ) = 0
for q ? 0. The module A \Gammaq;2q (K) is generated by monomials u J , #J = q. Since
d(u i ) = v i , it follows easily that there are no non­zero cocycles in A \Gammaq;2q (K). Hence,
H \Gammaq;2q (ZK ) = 0.
The assertion (c) follows from (e) and (49).
It also follows from (e) that H 3 (ZK ) = H \Gamma1;4 (ZK ). The basis for A \Gamma1;4 (K)
consists of monomials u j v i , i 6= j. We have d(u j v i ) = v i v j and d(u i u j ) = u j v i \Gamma u i v j .
It follows that u j v i is a cocycle if and only if fi; jg is not a 1­simplex in K; in this case
two cocycles u j v i and u i v j represent the same cohomology class. The assertion (d)
now follows easily.

64 VICTOR M. BUCHSTABER AND TARAS E. PANOV
0
2
4
. . .
2m
0
\Gamma1
\Delta \Delta \Delta
\Gamma(m \Gamma n)
\Gammam
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
(a) arbitrary K n\Gamma1
0
2
4
. . .
2m
0
\Gamma1
\Delta \Delta \Delta
\Gamma(m \Gamma n)
\Gammam
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
(b) jKj = S n\Gamma1
Figure 8. Possible locations of non­zero bigraded Betti numbers
b \Gammaq;2p (ZK ) (marked by \Lambda).
The remaining assertion (f) follows from the fact that the monomial u I v J 2
A \Lambda (K) of maximal total degree m + n necessarily has #I + #J = m, #J = n,
#I = m \Gamma n.
Lemma 4.2.7 shows that non­zero bigraded Betti numbers b r;2p (ZK ), r 6= 0
appear only in the ``strip'' bounded by the lines p = m, r = \Gamma1, p + r = 1 and
p + r = n in the second quadrant (see Figure 8 (a)).
The homogeneous component C \Gammaq;2p (ZK ) has the basis consisting of cellular
chains D I T J with I 2 K, #I = p \Gamma q, #J = q. It follows that
dimC \Gammaq;2p (ZK ) = f p\Gammaq\Gamma1
\Gamma m\Gammap+q
q
\Delta
(50)
(we assume
\Gamma i
j
\Delta = 0 if i ! j or j ! 0), where (f 0 ; f 1 ; : : : ; fn\Gamma1 ) is the f­vector of
K n\Gamma1 and f \Gamma1 = 1. The differential @ c does not change the second degree, i.e.
@ c : C \Gammaq;2p (ZK ) ! C \Gammaq\Gamma1;2p (ZK ):
Hence, the chain complex C \Lambda;\Lambda (ZK ) splits as follows:
[C \Lambda;\Lambda (ZK ); @ c ] =
m
M
p=0
[C \Lambda;2p (ZK ); @ c ]:
Remark. The similar decomposition holds also for the cellular cochain complex
[C \Lambda;\Lambda (ZK ); d c ] ¸ = [A \Lambda;\Lambda (K); d].
Let us consider the Euler characteristic of the complex [C \Lambda;2p (ZK ); @ c ]:
ü p (ZK ) :=
m
X
q=0
(\Gamma1) q dimC \Gammaq;2p (ZK ) =
m
X
q=0
(\Gamma1) q b \Gammaq;2p (ZK ):
(51)
Define the generating polynomial ü(ZK ; t) as
ü(ZK ; t) =
m
X
p=0
ü p (ZK )t 2p :
The following theorem calculates this polynomial in terms of the h­vector of K.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 65
Theorem 4.2.8. For any (n\Gamma1)­dimensional simplicial complex K with m vertices
holds
ü(ZK ; t) = (1 \Gamma t 2 ) m\Gamman (h 0 + h 1 t 2 + \Delta \Delta \Delta + hn t 2n );
(52)
where (h 0 ; h 1 ; : : : ; hn ) is the h­vector of K.
Proof. It follows from (51) and (50) that
ü p (ZK ) =
m
X
j=0
(\Gamma1) p\Gammaj f j \Gamma1
` m \Gamma j
p \Gamma j
'
;
(53)
Then
(54) ü(ZK ; t) =
m
X
p=0
ü p (K)t 2p =
m
X
p=0
m
X
j=0
t 2j t 2(p\Gammaj) (\Gamma1) p\Gammaj f j \Gamma1
` m \Gamma j
p \Gamma j
'
=
m
X
j=0
f j \Gamma1 t 2j (1 \Gamma t 2 ) m\Gammaj = (1 \Gamma t 2 ) m
n
X
j=0
f j \Gamma1 (t \Gamma2 \Gamma 1) \Gammaj :
Denote h(t) = h 0 + h 1 t + \Delta \Delta \Delta + hn t n . Then it follows from (7) that
t n h(t \Gamma1 ) = (t \Gamma 1) n
n
X
i=0
f i\Gamma1 (t \Gamma 1) \Gammai :
Substituting t \Gamma2 for t above, we finally obtain from (54)
ü(ZK ; t)
(1 \Gamma t 2 ) m = t \Gamma2n h(t 2 )
(t \Gamma2 \Gamma 1) n = h(t 2 )
(1 \Gamma t 2 ) n ;
which is equivalent to (52).
Theorem 4.2.8 allows to express the numbers of faces of a simplicial complex
in terms of the bigraded Betti numbers of the corresponding moment­angle com­
plex ZK .
Corollary 4.2.9. For any simplicial complex K the Euler characteristic of the
corresponding moment­angle complex ZK is zero.
Proof. We have
ü(ZK ) =
m
X
p;q=0
(\Gamma1) \Gammaq+2p b \Gammaq;2p (ZK ) =
m
X
p=0
ü p (ZK ) = ü(ZK ; 1)
Now the statement follows from (52).
Remark. Another proof of the above corollary follows from the observation that
the diagonal subgroup S 1 ae T m always acts freely on ZK (see also section 4.4).
Hence, there exists a principal S 1 ­bundle ZK ! ZK =S 1 , which implies ü(ZK ) = 0.
The torus T m = ae \Gamma1 (1; : : : ; 1) is a cellular subcomplex of ZK (see Lemma 3.3.3).
The cellular cochain subcomplex C \Lambda (T m ) ae C \Lambda (ZK ) ¸ = A \Lambda (K) has the basis consist­
ing of cochains (T I ) \Lambda and is mapped to the exterior algebra \Lambda[u 1 ; : : : ; um ] ae A \Lambda (K)
under the isomorphism of Theorem 4.2.6. It follows that there is an isomorphism
of modules
C \Lambda (ZK ; T m ) ¸ = A \Lambda (K)=\Lambda[u 1 ; : : : ; um ]:
(55)

66 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Likewise, we introduce relative bigraded Betti numbers
b \Gammaq;2p (ZK ; T m ) = dimH \Gammaq;2p \Theta C \Lambda (ZK ; T m ); d
\Lambda ; q; p = 0; : : : ; m;
(56)
define the p­th relative Euler characteristic ü p (ZK ; T m ) as
ü p (ZK ; T m ) =
m
X
q=0
(\Gamma1) q dimC \Gammaq;2p (ZK ; T m ) =
m
X
q=0
(\Gamma1) q b \Gammaq;2p (ZK ; T m );
(57)
and define the generating polynomial ü(ZK ; T m ; t) as
ü(ZK ; T m ; t) =
m
X
p=0
ü p (ZK ; T m )t 2p :
Theorem 4.2.10. For any (n \Gamma 1)­dimensional simplicial complex K with m ver­
tices holds
ü(ZK ; T m ; t) = (1 \Gamma t 2 ) m\Gamman (h 0 + h 1 t 2 + \Delta \Delta \Delta + hn t 2n ) \Gamma (1 \Gamma t 2 ) m :
(58)
Proof. Since C \Lambda (T m ) = \Lambda[u 1 ; : : : ; um ] and bideg u i = (\Gamma1; 2), we have
dimC \Gammaq (T m ) = dimC \Gammaq;2q (T m ) =
\Gamma m
q
\Delta
:
Combining (55), (51) and (57) we get
ü p (ZK ; T m ) = ü p (ZK ) \Gamma (\Gamma1) p dimC \Gammap;2p (T m ):
Hence,
ü(ZK ; T m ; t) = ü(ZK ; t) \Gamma
m
X
p=0
(\Gamma1) p
\Gamma m
p
\Delta
t 2p
= (1 \Gamma t 2 ) m\Gamman (h 0 + h 1 t 2 + \Delta \Delta \Delta + hn t 2n ) \Gamma (1 \Gamma t 2 ) m ;
where we used (52).
We will use Theorem 4.2.10 in section 4.5.
Theorem 4.2.11. Suppose that K n\Gamma1 is Cohen--Macaulay, and let J be the ideal
in k(K) generated by a degree­two regular sequence of length n. Then the following
isomorphism of algebras holds:
H \Lambda (ZK ) ¸ = Tor k[v1 ;::: ;vm ]=J
\Gamma k(K)=J ; k
\Delta :
Proof. This follows from Theorem 4.2.1 and Lemma 1.4.9.
Note that the k­algebra k(K)=J is finite­dimensional. This fact sometimes makes
Theorem 4.2.11 more convenient for calculations (in the Cohen--Macaulay case)
than general Theorem 4.2.1.
4.3. Cohomology of moment­angle complex ZK : the case of spherical K.
If K is a simplicial sphere, then the complex ZK is a manifold (Lemma 3.2.2).
This imposes additional conditions on the cohomology of ZK ; the corresponding
results are described in this section together with some interesting interpretations
of combinatorial problems reviewed in chapter 1.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 67
Theorem 4.3.1. Let K be an (n \Gamma 1)­dimensional simplicial sphere, and ZK the
corresponding moment­angle manifold, dimZK = m + n. Then the fundamental
cohomology class of ZK is represented by any monomial \Sigmav I u J 2 A \Lambda (K) of bidegree
(\Gamma(m \Gamma n); 2m) such that I is an (n \Gamma 1)­simplex of K and I `` J = ;. The sign
depends on the orientation of ZK .
Proof. It follows from Lemma 4.2.7 (f) that H m+n (ZK ) = H \Gamma(m\Gamman);2m (ZK ). By
definition, the module A \Gamma(m\Gamman);2m (K) is spanned by monomials v I u J such that
I 2 K n\Gamma1 , #I = n, J = [m]n I. Any such monomial is a cocycle. Suppose that I; I 0
are two (n \Gamma 1)­simplices of K n\Gamma1 that share a common (n \Gamma 2)­face. We claim that
the corresponding cocycles v I u J , v I 0 u J 0 , where J = [m] n I, J 0 = [m] n I 0 , represent
the same cohomology class (up to a sign). Indeed, let v I u J = v i 1
\Delta \Delta \Delta v i n u j1 \Delta \Delta \Delta u jm\Gamman ,
v I 0 u J 0 = v i 1
\Delta \Delta \Delta v i n\Gamma1 v j1 u i n u j2 \Delta \Delta \Delta u jm\Gamman . Since any (n \Gamma 2)­face of K is contained in
exactly two (n \Gamma 1)­faces, the identity
d(v i 1
\Delta \Delta \Delta v i n\Gamma1 u i n u j1 u j2 \Delta \Delta \Delta u jm\Gamman )
= v i 1 \Delta \Delta \Delta v i n u j1 \Delta \Delta \Delta u jm\Gamman \Gamma v i 1 \Delta \Delta \Delta v i n\Gamma1 v j1 u i n u j2 \Delta \Delta \Delta u jm\Gamman
holds in A \Lambda (K) ae \Lambda[u 1 ; : : : ; um
]\Omega k(K). Hence, [v I u J ] = [v I 0 u J 0 ] (as cohomol­
ogy classes). Since K n\Gamma1 is a simplicial sphere, any two (n \Gamma 1)­simplices can be
connected by a chain of simplices such that any two successive simplices share a
common (n \Gamma 2)­face. Thus, all monomials v I u J in A \Gamma(m\Gamman);2m (K) represent the
same cohomology class (up to a sign). This class is the generator of H m+n (ZK ),
i.e. the fundamental cohomology class of ZK .
Remark. In the proof of the above theorem we have used two combinatorial prop­
erties of K n\Gamma1 . The first one is that any (n \Gamma 2)­face is contained in exactly two
(n \Gamma 1)­faces, and the second is that any two (n \Gamma 1)­simplices can be connected by
a chain of simplices such that any two successive simplices share a common (n \Gamma 2)­
face. Both properties hold for any simplicial manifold. Hence, for any simplicial
manifold K n\Gamma1 we have b m+n (ZK ) = b \Gamma(m\Gamman);2m (ZK ) = 1, and the generator of
H m+n (ZK ) can be chosen as described in Theorem 4.3.1.
Corollary 4.3.2. The Poincar'e duality for the moment angle manifold ZK defined
by a simplicial sphere K n\Gamma1 regards the bigraded structure in the (co)homology, i.e.
H \Gammaq;2p (ZK ) ¸ = H \Gamma(m\Gamman)+q;2(m\Gammap) (ZK ):
In particular,
b \Gammaq;2p (ZK ) = b \Gamma(m\Gamman)+q;2(m\Gammap) (ZK ): \Lambda
(59)
Corollary 4.3.3. Let K n\Gamma1 be an (n \Gamma 1)­dimensional simplicial sphere, and ZK
the corresponding moment­angle complex, dimZK = m+ n. Then
(a) b \Gammaq;2p (ZK ) = 0 for q – m \Gamma n, with only exception b \Gamma(m\Gamman);2m = 1;
(b) b \Gammaq;2p (ZK ) = 0 for p \Gamma q – n, with only exception b \Gamma(m\Gamman);2m = 1.
It follows that if K n\Gamma1 is a simplicial sphere, then non­zero bigraded Betti num­
bers b r;2p (ZK ), r 6= 0, r 6= m \Gamma n, appear only in the ``strip'' bounded by the lines
r = \Gamma(m \Gamma n \Gamma 1), r = \Gamma1, p + r = 1 and p + r = n \Gamma 1 in the second quadrant
(see Figure 8 (b)). Compare this with Figure 8 (a) corresponding to the case of
general K.

68 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Example 4.3.4. Let K = @ \Delta m\Gamma1 . Then k(K) = k[v 1 ; : : : ; vm ]=(v 1 \Delta \Delta \Delta vm ) (Exam­
ple 1.3.4). It is easy to see that the cohomology groups
H[k(K)\Omega \Lambda[u 1 ; : : : ; um ]; d]
(see Theorem 4.2.2) are generated by the classes 1 and [v 1 v 2 \Delta \Delta \Delta v m\Gamma1 um ]. We have
deg(v 1 v 2 \Delta \Delta \Delta v m\Gamma1 um ) = 2m \Gamma 1, and Theorem 4.3.1 shows that v 1 v 2 \Delta \Delta \Delta v m\Gamma1 um
represents the fundamental cohomology class of ZK ¸ = S 2m\Gamma1 (see Example 3.1.7).
Example 4.3.5. Let K be the boundary complex of an m­gon P 2 with m – 4. We
have k(K) = k[v 1 ; : : : ; v m ]=I P , where IP is generated by the monomials v i v j , i\Gammaj 6=
0; 1 mod m. The complex ZK = ZP is a smooth manifold of dimension m+ 2. The
Betti numbers and the cohomology rings of these manifolds were calculated in [20].
Namely,
dimH k (ZP ) =
8 ? !
? :
1 for k = 0; m+ 2;
0 for k = 1; 2; m;m+ 1;
(m \Gamma 2)
\Gamma m\Gamma2
k\Gamma2
\Delta \Gamma
\Gamma m\Gamma2
k\Gamma1
\Delta \Gamma
\Gamma m\Gamma2
k\Gamma3
\Delta for 3 Ÿ k Ÿ m \Gamma 1:
For example, in the case m = 5 the group H 3 (ZP ) has 5 generators represented by
the cocycles v i u i+2 2
k(K)\Omega \Lambda[u 1 ; : : : ; u 5 ], i = 1; : : : ; 5, while the group H 4 (ZP )
has 5 generators represented by the cocycles v j u j+2 u j+3 , j = 1; : : : ; 5. As it follows
from Theorem 4.3.1, the product of cocycles v i u i+2 and v j u j+2 u j+3 represents a
non­zero cohomology class in H 7 (ZP ) if and only if all indices i; i + 2; j; j + 2; j + 3
are different. Thus, for each of the 5 cohomology classes [v i u i+2 ] there is the unique
(Poincar'e dual) cohomology class [v j u j+2 u j+3 ] such that the product [v i u i+2 ] \Delta
[v j u j+2 u j+3 ] is non­zero.
It follows from (51) and (59) that for any simplicial sphere K holds
ü p (ZK ) = (\Gamma1) m\Gamman üm\Gammap (ZK ):
From this and (52) we get
h 0 + h 1 t 2 + \Delta \Delta \Delta + hn t 2n
(1 \Gamma t 2 ) n = (\Gamma1) m\Gamman üm + üm\Gamma1 t 2 + \Delta \Delta \Delta + ü 0 t 2m
(1 \Gamma t 2 ) m
= (\Gamma1) n ü 0 + ü 1 t \Gamma2 + \Delta \Delta \Delta + üm t \Gamma2m
(1 \Gamma t \Gamma2 ) m = (\Gamma1) n h 0 + h 1 t \Gamma2 + \Delta \Delta \Delta + hn t \Gamma2n
(1 \Gamma t \Gamma2 ) n
= h 0 t 2n + h 1 t 2(n\Gamma1) + \Delta \Delta \Delta + hn
(1 \Gamma t 2 ) n :
Hence, h i = hn\Gammai . Thus, the Dehn--Sommerville equations are a corollary of the
bigraded Poincar'e duality (59).
The identity (52) also allows to interpret different inequalities for the f­vectors
of simplicial spheres (respectively, simplicial manifolds) in terms of topological in­
variants (bigraded Betti numbers) of the corresponding moment­angle manifolds
(respectively, complexes) ZK .
Example 4.3.6. It follows from Lemma 4.2.7 that for any K we have
ü 0 (ZK ) = 1; ü 1 (ZK ) = 0;
ü 2 (ZK ) = \Gammab \Gamma1;4 (ZK ) = \Gammab 3 (ZK ); ü 3 (ZK ) = b \Gamma2;6 (ZK ) \Gamma b \Gamma1;6 (ZK )

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 69
(note that b 4 (ZK ) = b \Gamma2;6 (ZK ), while b 5 (ZK ) = b \Gamma1;6 (ZK ) + b \Gamma3;8 (ZK )). Now,
identity (52) shows that
h 0 = 1;
h 1 = m \Gamma n;
h 2 =
\Gamma m\Gamman+1
2
\Delta \Gamma b 3 (ZK );
h 3 =
\Gamma m\Gamman+2
3
\Delta \Gamma (m \Gamma n)b \Gamma1;4 (ZK ) + b \Gamma2;6 (ZK ) \Gamma b \Gamma1;6 (ZK ):
It follows that the inequality h 1 Ÿ h 2 (n – 4) from the Generalized Lower Bound
hypothesis (13) for simplicial spheres is equivalent to the following:
b 3 (ZK ) Ÿ
\Gamma m\Gamman
2
\Delta :
(60)
The next inequality h 2 Ÿ h 3 (n – 6) from (13) is equivalent to the following
inequality for the bigraded Betti numbers of ZK :
\Gamma m\Gamman+1
3
\Delta \Gamma (m \Gamma n \Gamma 1)b \Gamma1;4 (ZK ) + b \Gamma2;6 (ZK ) \Gamma b \Gamma1;6 (ZK ) – 0:
(61)
We see that the combinatorial Generalized Lower Bound inequalities are in­
terpreted as ``topological'' inequalities for the (bigraded) Betti numbers of a cer­
tain manifold. So, one can try to use topological methods (such as the equivariant
topology or Morse theory) to prove inequalities like (60) or (61). Such topological
approach to problems like g­conjecture or Generalized Lower Bound has the ad­
vantage of being independent on whether the simplicial sphere K is polytopal or
not. Indeed, all known proofs of the necessity of g­theorem for simplicial polytopes
(including the original one by Stanley given in section 2.1, McMullen's proof [59],
and the recent proof by Timorin [85]) follow the same scheme. Namely, the num­
bers h i , i = 1; : : : ; n, are interpreted as the dimensions of graded components A i
of a certain graded algebra A satisfying the Hard Lefschetz Theorem. The latter
means that there is an element ! 2 A 1 such that the multiplication by ! defines
a monomorphism A i ! A i+1 for i !
\Theta n
2
\Lambda
. This implies h i Ÿ h i+1 for i !
\Theta n
2
\Lambda
(see
section 2.1). However, such an element ! is lacking for non­polytopal K, which
means that a new technique has to be developed for proving the g­conjecture for
simplicial spheres.
As it was mentioned in section 1.3, simplicial spheres are Gorenstein* complexes.
Using theorems 1.4.12, 1.4.13 and our Theorem 4.2.1 we obtain the following solu­
tion of the analogue of Problem 3.2.3.
Theorem 4.3.7. The complex ZK is a Poincar'e duality complex (over k) if and
only if for any simplex I 2 K (including I = ;) the subcomplex linkI has the
homology of a sphere of dimension dim(link I).
4.4. Partial quotients of manifold ZP . Here we return to the case of polytopal
K (i.e. K = KP ) and study quotients of ZP by freely acting subgroups H ae T m .
For any combinatorial simple polytope P n denote by s(P n ) the maximal dimen­
sion of subgroups H ae T m that act freely on ZP . The number s(P n ) is obviously
a combinatorial invariant of P n .
Problem 4.4.1 (V. M. Buchstaber). Express s(P n ) via known combinatorial in­
variants of P n .
Proposition 4.4.2. If P n has m facets, then s(P n ) Ÿ m \Gamma n.

70 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Proof. Every subgroup of T m of dimension ? m \Gamma n intersects non­trivially with
any n­dimensional isotropy subgroup, and therefore can not act freely on ZP .
Proposition 4.4.3. The diagonal circle subgroup S d := f(e 2úi' ; : : : ; e 2úi' ) 2
T m g, ' 2 R, acts freely on any ZP . Thus, s(P n ) – 1.
Proof. Since any isotropy subgroup for ZP is coordinate (see Definition 3.1.1), it
intersects with S d only at the unit.
Another lower bound for the number s(P n ) was proposed in [48]. Let F =
fF 1 ; : : : ; Fmg be the set of facets of P n . The surjective map % : F ! [k] (where
[k] = f1; : : : ; kg) is called the regular k­paint colouring of facets of P n if %(F i ) 6=
%(F j ) whenever F i `` F j 6= ;. The chromatic number fl(P n ) of a polytope P n is the
minimal k for which there exists a regular k­paint colouring of facets of P n .
Example 4.4.4. Suppose P n is a 2­neighbourly simple polytope with m facets.
Then fl(P n ) = m.
Proposition 4.4.5 ([48]). The following inequality holds:
s(P n ) – m \Gamma fl(P n ):
Proof. The map % : F ! [k] defines the epimorphism of tori ~
% : T m ! T k . It is easy
to see that if % is a regular colouring, then Ker ~
% ¸ = T m\Gammak acts freely on ZP .
Let H ae T m be a subgroup of dimension r Ÿ m \Gamma n. Choosing a basis in H, we
can write it in the form
H =
\Phi (e 2úi(s11 '1+\Delta\Delta\Delta+s 1r 'r ) ; : : : ; e 2úi(sm1'1+\Delta\Delta\Delta+s mr'r ) ) 2 T m
\Psi
;
(62)
where ' i 2 R, i = 1; : : : ; r. The m \Theta r integer matrix S = (s ij ) defines a
monomorphism Z r !Z m whose image is a direct summand in Z m . For any subset
fi 1 ; : : : ; i n g ae [m] denote by S “ i 1 ;::: ; “ i n the (m \Gamma n) \Theta r submatrix of S obtained by
deleting the rows i 1 ; : : : ; i n . Recall that any vertex v 2 P n is the intersection of
n facets (see (30)). The following criterion of freeness for the action of H on ZP
holds.
Lemma 4.4.6. The subgroup (62) acts freely on ZP if and only if for any vertex
v = F i 1
``: : :``F i n of P n the (m\Gamman)\Thetar­submatrix S “ i 1 ;::: ; “ i n defines the monomorphism
Z r ,! Z m\Gamman to a direct summand.
Proof. It follows from Definition 3.1.1 that the orbits of the T m ­action on ZP
corresponding to the vertices of P n have maximal (rank n) isotropy subgroups. The
isotropy subgroup corresponding to the vertex v = F i 1
`` : : : `` F i n is the coordinate
subtorus T n
i 1 ;::: ;i n
ae T m . The subgroup (62) acts freely on ZP if and only if it
intersects each isotropy subgroup only at the unit. This is equivalent to the condition
that the map H \Theta T n
i 1 ;::: ;i n
! T m is injective for any v = F i 1
`` : : : `` F i n . The latter
map is injective whenever the image of the corresponding map Z r+n ,! Z m is a
direct summand of Z m . The matrix of the map Z r+n ,! Z m is obtained by adding
n columns (0; : : : ; 0; 1; 0; : : : ; 0) t (1 stands at the place i j , j = 1; : : : ; n) to S. This
matrix defines a direct summand of Z m exactly when the same is true for each
S “ i 1 ;::: ; “ i n
.
In particular, for subgroups of rank m \Gamma n we get

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 71
Corollary 4.4.7. The subgroup (62) of rank r = m \Gamma n acts freely on ZP if
and only if for any vertex v = F i 1 `` : : : `` F i n of P n the minor S “ i 1 ;::: ; “ i n
satisfies
det S “ i 1 ::: “ i n
= \Sigma1.
Proposition 4.4.8. A simple polytope P n admits a characteristic map if and only
if s(P n ) = m \Gamma n.
Proof. Proposition 3.1.5 shows that if P n admits a characteristic map `, then the
(m \Gamma n)­dimensional subgroup H(`) acts freely on ZP , whence s(P n ) = m \Gamma n. Now
suppose s(P n ) = m \Gamma n, i.e. there exists a subgroup (62) of rank r = m \Gamma n that acts
freely on ZP . The corresponding m \Theta (m \Gamma n)­matrix S defines a monomorphism
Z m\Gamman !Z m whose image is a direct summand. It follows that there is an n \Theta m­
matrix \Lambda such that the sequence
0 \Gamma\Gamma\Gamma\Gamma! Z m\Gamman S
\Gamma\Gamma\Gamma\Gamma! Z m \Lambda
\Gamma\Gamma\Gamma\Gamma! Z n \Gamma\Gamma\Gamma\Gamma! 0
is exact. Since S satisfies the condition of Corollary 4.4.7, the matrix \Lambda satisfies the
condition (26), thus defining a characteristic map for P n .
For any subgroup (62) of dimension r = m \Gamma n define the following linear forms
in k[v 1 ; : : : ; v m ]:
w i = s 1i v 1 + \Delta \Delta \Delta + s mi vm ; i = 1; : : : ; m \Gamma n:
(63)
Suppose M 2n is a quasitoric manifold over P n with characteristic map `. Write
the subgroup H(`) in the form (62); this defines elements (63). Under these as­
sumptions the following statement holds.
Lemma 4.4.9. The following isomorphism of algebras holds:
H \Lambda (ZP ) ¸ = Tor k[w1 ;::: ;wm\Gamman ]
\Gamma
H \Lambda (M 2n ); k
\Delta
;
where the k[w 1 ; : : : ; wm\Gamman ]­module structure on H \Lambda (M 2n ) = k[v 1 ; : : : ; v m ]=I P +J `
is defined by (63).
Proof. Theorem 4.2.11 shows that
H \Lambda (ZK ) ¸ = Tor k[v1 ;::: ;vm ]=J `
\Gamma k(K)=J ` ; k
\Delta :
The quotient k[v 1 ; : : : ; vm ]=J ` is identified with k[w 1 ; : : : ; wm\Gamman ].
Theorem 4.4.10. The Leray--Serre spectral sequence of the T m\Gamman ­bundle ZP !
M 2n collapses at the E 3 term. Furthermore, the following isomorphism of algebras
holds:
H \Lambda (ZP ) ¸ = H
\Theta \Lambda[u 1 ; : : : ; um\Gamman
]\Omega (k(P )=J ` ); d
\Lambda ;
bideg v i = (0; 2); bideg u i = (\Gamma1; 2);
d(u i ) = w i ; d(v i ) = 0:
Proof. Since H \Lambda (T m\Gamman ) = \Lambda[u 1 ; : : : ; um\Gamman ] and H \Lambda (M 2n ) = k(P )=J ` , we have
E 3
¸ = H
\Theta (k(P )=J `
)\Omega \Lambda[u 1 ; : : : ; um\Gamman ]; d
\Lambda
:
By Lemma 1.4.6,
H
\Theta (k(P )=J `
)\Omega \Lambda[u 1 ; : : : ; um\Gamman ]; d
\Lambda ¸ = Tor k[w1 ;::: ;wm\Gamman ]
\Gamma H \Lambda (M 2n ); k
\Delta :
Combining the above two identities with Lemma 4.4.9 we get E 3 = H \Lambda (ZP ), which
concludes the proof.

72 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Now we are going to describe the cohomology of the quotient ZP =H for arbitrary
freely acting subgroup H. First, we write H in the form (62) and choose an (m \Gamma
r) \Theta m­matrix T = (t ij ) of rank (m \Gamma r) satisfying T \Delta S = 0. This is done in the
same way as in the proof of Proposition 4.4.8 (in particular, T is the characteristic
matrix for the quasitoric manifold ZP =H in the case r = m \Gamma n).
Theorem 4.4.11. The following isomorphism of algebras holds:
H \Lambda (ZP =H) ¸ = Tor k[t1 ;::: ;t m\Gammar ]
\Gamma k(P ); k
\Delta ;
where the k[t 1 ; : : : ; t m\Gammar ]­module structure on k(P ) = k[v 1 ; : : : ; v m ]=I P is given by
the map
k[t 1 ; : : : ; t m\Gammar ] ! k[v 1 ; : : : ; vm ]
t i ! t i1 v 1 + \Delta \Delta \Delta + t im vm :
Remark. Theorem 4.4.11 reduces to Theorem 4.2.1 in the case r = 0 and to Exam­
ple 4.1.5 in the case r = m \Gamma n.
Proof of Theorem 4.4.11. The inclusion of the subgroup T r ¸ = H ,! T m defines the
map of classifying spaces h : BT r ! BT m . Let us consider the commutative square
E \Gamma\Gamma\Gamma\Gamma! BT P
? ? y
? ? y p
BT r h
\Gamma\Gamma\Gamma\Gamma! BT m ;
where the left vertical arrow is the pullback along h. It can be easily seen that
E is homotopy equivalent to the quotient ZP =H. The Eilenberg--Moore spectral
sequence of the above square converges to the cohomology of ZP =H and has
E 2 = Tor k[v1 ;::: ;vm ]
\Gamma k(P ); k[w 1 ; : : : ; w r ]
\Delta
;
where the k[v 1 ; : : : ; v m ]­module structure on k[w 1 ; : : : ; w r ] is defined by the ma­
trix S, i.e. by the map v i ! s i1 w 1 + : : : + s ir w r . It can be shown in the same way
as in the proof of Theorem 4.2.1 (using cellular decompositions) that the spectral
sequence collapses at the E 2 term and the following isomorphism of algebras holds:
H \Lambda (ZP =H) = Tor k[v1 ;::: ;vm ]
\Gamma k(P ); k[w 1 ; : : : ; w r ]
\Delta
:
(64)
Now put \Lambda = k[v 1 ; : : : ; vm ], \Gamma = k[t 1 ; : : : ; t m\Gammar ], A = k[w 1 ; : : : ; w r ], and C = k(P )
in Theorem 1.4.10. Since \Lambda here is a free \Gamma­module
and\Omega = \Lambda=\Gamma ¸ = k[w 1 ; : : : ; w r ],
the spectral sequence f e
E s ; e
d s g arises. Its E 2 term is
e
E 2 = Tor k[w1 ;::: ;wr ]
i
k[w 1 ; : : : ; w r ]; Tor k[t1 ;::: ;t m\Gammar ]
\Gamma k(P ); k
\Delta j
;
and it converges to Tor k[v1 ;::: ;vm ] (k(P ); k[w 1 ; : : : ; w r ]). Since k[w 1 ; : : : ; w r ] is a free
k[w 1 ; : : : ; w r ]­module, we have
e
E p;q
2 = 0 for p 6= 0; e
E 0;\Lambda
2 = Tor k[t1 ;::: ;t m\Gammar ]
\Gamma k(P ); k
\Delta :
Thus, the spectral sequence collapses at the E 2 term, and the following isomorphism
of algebras holds:
Tor k[v1 ;::: ;vm ]
\Gamma k(P ); k[w 1 ; : : : ; w r ]
\Delta ¸ = Tor k[t1 ;::: ;t m\Gammar ]
\Gamma k(P ); k
\Delta
;
which together with (64) proves the theorem.
Corollary 4.4.12. H \Lambda (ZP =H) ¸ = H
\Theta
\Lambda[u 1 ; : : : ; um\Gammar
]\Omega k; d
\Lambda
, where du i = (t i1 v 1 +
: : : + t im v m ), dv i = 0, bideg v i = (0; 2), bideg u i = (\Gamma1; 2).

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 73
Example 4.4.13. Let H = S d (see Proposition 4.4.3). In this situation the matrix
S is the column of m units. By Theorem 4.4.11,
H \Lambda (ZP =S d ) ¸ = Tor k[t1 ;::: ;t m\Gamma1 ]
\Gamma k(P ); k
\Delta ;
(65)
where the k[t 1 ; : : : ; t m\Gamma1 ]­module structure on k(P ) = k[v 1 ; : : : ; vm ]=I is defined
by
t i \Gamma! v i \Gamma v m ; i = 1; : : : ; m \Gamma 1:
Suppose that the S 1 ­bundle ZP ! ZP =S d is classified by the map c : ZP =S d !
BT 1 ¸ = CP 1 . Since H \Lambda (C P 1 ) = k[w], the element c \Lambda (w) 2 H 2 (ZP =S d ) is defined.
Lemma 4.4.14. P n is q­neighbourly if and only if (c \Lambda (w)) q 6= 0.
Proof. The map c \Lambda takes the cohomology ring H \Lambda (BT 1 ) ¸ = k[w] to the subring
k(P
)\Omega k[t1 ;::: ;t m\Gamma1 ] k = Tor 0
k[t1 ;::: ;t m\Gamma1 ] (k(P ); k) of H \Lambda (ZP =H). This subring is iso­
morphic to the quotient k(P )=(v 1 = \Delta \Delta \Delta = vm ). Now the assertion follows from the
fact that a polytope P n is q­neighbourly if and only if the ideal IP does not contain
monomials of degree ! q + 1.
4.5. Bigraded Poincar'e duality and analogues of Dehn--Sommerville
equations for simplicial manifolds. Here we assume that K n\Gamma1 is a simplicial
manifold. In this case the moment­angle complex ZK is not a manifold, however, its
singularities can be easily treated. Indeed, the cubical complex cc(K) (Construc­
tion 1.5.9) is homeomorphic to j cone(K)j, and the vertex of the cone is the point
p = (1; : : : ; 1) 2 cc(K) ae I m . Let U '' (p) ae cc(K) be a small neighbourhood of p
in cc(K). Then the closure of U '' (p) is also homeomorphic to j cone(K)j. It follows
from the definition of ZK (see (39)) that U '' (T m ) := ae \Gamma1 (U '' (p)) ae ZK is a small
invariant neighbourhood of the torus T m = ae \Gamma1 (p) in ZK . For small '' the closure
of U '' (T m ) is homeomorphic to j cone(K)j \Theta T m . Removing U '' (T m ) from ZK we
obtain a manifold with boundary, which we denote WK . Thus, we have
WK = ZK n U '' (T m ); @WK = jKj \Theta T m :
Note that since the neighbourhood U '' (T m ) is T m ­stable, the torus T m acts on WK .
Theorem 4.5.1. The manifold with boundary WK is equivariantly homotopy
equivalent to the moment­angle complex WK (see (39)). There is a canonical rela­
tive homeomorphism of pairs (WK ; @WK ) ! (ZK ; T m ).
Proof. To prove the first assertion we construct homotopy equivalence cc(K) n
U '' (p) ! cub(K) as it is shown on Figure 9. This map is covered by an equivariant
homotopy equivalence WK = ZK n U '' (T m ) ! WK . The second assertion follows
easily from the definition of WK .
According to Lemma 3.3.1, the moment­angle complex WK ae (D 2 ) m has a
cellular structure with 5 different cell types D i , I i , 0 i , T i , 1 i , i = 1; : : : ; m (see
Figure 7). The homology of WK (and therefore of WK ) can be calculated by means
of the corresponding cellular chain complex, which we denote [C \Lambda (WK ); @ c ]. Though
WK has more types of cells than ZK (recall that ZK has only 3 cell types D i , T i , 1 i ),
the cellular chain complex [C \Lambda (WK ); @ c ] can be canonically made bigraded as well.
Namely, the following statement holds (compare with (46)).

74 VICTOR M. BUCHSTABER AND TARAS E. PANOV
u
u
u
u
u
u
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma \Gamma`
\Gamma
\Gamma
oe
?
j
j
j
j
j
j
\Omega \Omega \Omega \Omega \Omega \Omega
j
j
j
j+
\Omega \Omega \Omega \Omega AE
X
X
X
X
X J
J J
X
X
X
Xy J
J]
C
C
C
C C
@
@
C
C
C CW
@ @R
ae ¸¸¸¸¸ cc(K) n U '' (p)
J
J
J
¸
¸
¸
cub(K)
Figure 9. Homotopy equivalence cc(K) n U '' (p) ! cub(K).
Lemma 4.5.2. Put
bideg D i = (0; 2); bideg T i = (\Gamma1; 2); bideg I i = (1; 0);
(66)
bideg 0 i = bideg 1 i = (0; 0); i = 1; : : : ; m:
This makes the cellular chain complex [C \Lambda (WK ); @ c ] a bigraded differential module
with differential @ c adding (\Gamma1; 0) to bidegree. The original grading of C \Lambda (WK ) by
dimensions of cells corresponds to the total degree (i.e. the dimension of a cell equals
the sum of its two degrees).
Proof. We need only to check that the differential @ c adds (\Gamma1; 0) to bidegree. This
follows from (66) and the following formulae:
@ c D i = T i ; @ c I i = 1 i \Gamma 0 i ; @ c T i = @ c 1 i = @ c 0 i = 0:
Note that unlike bigraded structure in C \Lambda (ZK ) elements of C \Lambda;\Lambda (WK ) may have pos­
itive first degree (due to the positive first degree of I i ). Nevertheless, the differential
@ c does not change the second degree (as in the case of ZK ), which allows to split the
bigraded complex C \Lambda;\Lambda (WK ) into the sum of complexes C \Lambda;2p (WK ), p = 0; : : : ; m.
In the same way as we did for ZK and for the pair (ZK ; T m ) we define
b q;2p (WK ) = dimH q;2p
\Theta C \Lambda;\Lambda (WK ); @ c
\Lambda ; \Gammam Ÿ q Ÿ m; 0 Ÿ p Ÿ m;
(67)
ü p (WK ) =
m
X
q=\Gammam
(\Gamma1) q dimC q;2p (WK ) =
m
X
q=\Gammam
(\Gamma1) q b q;2p (WK );
(68)
ü(WK ; t) =
m
X
p=0
ü p (WK )t 2p
(note that q above may be both positive and negative).
The following theorem provides the exact formula for the generating polynomial
ü(WK ; t) and is analogous to theorems 4.2.8 and 4.2.10.
Theorem 4.5.3. For any simplicial complex K n\Gamma1 with m vertices holds
ü(WK ; t) = (1 \Gamma t 2 ) m\Gamman (h 0 + h 1 t 2 + \Delta \Delta \Delta + hn t 2n ) +
\Gamma ü(K) \Gamma 1
\Delta (1 \Gamma t 2 ) m
= (1 \Gamma t 2 ) m\Gamman (h 0 + h 1 t 2 + \Delta \Delta \Delta + hn t 2n ) + (\Gamma1) n\Gamma1 hn (1 \Gamma t 2 ) m ;

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 75
where ü(K) = f 0 \Gamma f 1 + : : : + (\Gamma1) n\Gamma1 fn\Gamma1 = 1 + (\Gamma1) n\Gamma1 hn is the Euler charac­
teristic of K.
Proof. The definition of WK (see (39)) shows that D I I J 0LTP 1Q (see section 3.3)
is a cell of WK if and only if the following two conditions are satisfied:
(a) The set I [ J [ L is a simplex of K n\Gamma1 .
(b) #L – 1.
Let c ijlpq (WK ) denote the number of cells D I I J 0LTP 1Q ae WK with i = #I,
j = #J , l = #L, p = #P , q = #Q, i + j + l + p + q = m. It follows that
c ijlpq (WK ) = f i+j+l\Gamma1
\Gamma i+j+l
i
\Delta\Gamma j+l
l
\Delta\Gamma m\Gammai\Gammaj \Gammal
p
\Delta ;
(69)
where (f 0 ; : : : ; f n\Gamma1 ) is the f­vector of K (we also assume f \Gamma1 = 1 and f k = 0 for
k ! \Gamma1 or k ? n \Gamma 1). By (66),
bideg(D I I J 0LTP 1Q ) = (j \Gamma p; 2(i + p)):
Now we calculate ü r (WK ) using (68) and (69):
ü r (WK ) =
X
i;j;l;p
i+p=r;l–1
(\Gamma1) j \Gammap f i+j+l\Gamma1
\Gamma i+j+l
i
\Delta\Gamma j+l
l
\Delta\Gamma m\Gammai\Gammaj \Gammal
p
\Delta
:
Substituting s = i + j + l above we obtain
ü r (WK ) =
X
l;s;p
l–1
(\Gamma1) s\Gammar\Gammal f s\Gamma1
\Gamma s
r\Gammap
\Delta\Gamma s\Gammar+p
l
\Delta\Gamma m\Gammas
p
\Delta
=
X
s;p
i
(\Gamma1) s\Gammar f s\Gamma1
\Gamma s
r\Gammap
\Delta\Gamma m\Gammas
p
\Delta X
l–1
(\Gamma1) l
\Gamma s\Gammar+p
l
\Delta j
Since X
l–1
(\Gamma1) l
\Gamma s\Gammar+p
l
\Delta =
ae \Gamma1; s ? r \Gamma p;
0; s Ÿ r \Gamma p ;
we get
ü r (WK ) = \Gamma
X
s;p
s?r\Gammap
(\Gamma1) s\Gammar f s\Gamma1
\Gamma s
r\Gammap
\Delta\Gamma m\Gammas
p
\Delta
= \Gamma
X
s;p
(\Gamma1) r\Gammas f s\Gamma1
\Gamma s
r\Gammap
\Delta\Gamma m\Gammas
p
\Delta +
X
s
(\Gamma1) r\Gammas f s\Gamma1
\Gamma m\Gammas
r\Gammas
\Delta
:
The second sum in the above formula is exactly ü r (ZK ) (see (53)). To calculate
the first sum we observe that
P
p
\Gamma s
r\Gammap
\Delta \Gamma m\Gammas
p
\Delta =
\Gamma m
r
\Delta (this follows from calculating
the coefficient of ff r in both sides of the identity (1 + ff) s (1 + ff) m\Gammas = (1 + ff) m ).
Hence,
ü r (WK ) = \Gamma
X
s
(\Gamma1) r\Gammas f s\Gamma1
\Gamma m
r
\Delta
+ ü r (ZK ) = (\Gamma1) r
\Gamma m
r
\Delta\Gamma
ü(K) \Gamma 1
\Delta
+ ü r (ZK );
since \Gamma
P
s (\Gamma1) s f s\Gamma1 = ü(K) \Gamma 1 (remember that f \Gamma1 = 1). Finally, using (52) we
calculate
ü(WK ; t) =
m
X
r=0
ü r (WK )t 2r =
m
X
r=0
(\Gamma1) r
\Gamma m
r
\Delta\Gamma ü(K) \Gamma 1
\Delta t 2r +
m
X
r=0
ü r (ZK )t 2r
=
\Gamma ü(K) \Gamma 1
\Delta (1 \Gamma t 2 ) m + (1 \Gamma t 2 ) m\Gamman (h 0 + h 1 t 2 + \Delta \Delta \Delta + hn t 2n ):

76 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Suppose now that K is an orientable simplicial manifold. It is easy to see that
in this case WK is also orientable. Hence, there are relative Poincar'e duality iso­
morphisms:
H k (WK ) ¸ = H m+n\Gammak (WK ; @WK ); k = 0; : : : ; m:
(70)
Corollary 4.5.4 (Generalised Dehn--Sommerville equations). The following rela­
tions hold for the h­vector (h 0 ; h 1 ; : : : ; hn ) of any orientable simplicial mani­
fold K n\Gamma1 :
hn\Gammai \Gamma h i = (\Gamma1) i
\Gamma ü(K n\Gamma1 ) \Gamma ü(S n\Gamma1 )
\Delta\Gamma n
i
\Delta ; i = 0; 1; : : : ; n;
where ü(S n\Gamma1 ) = 1 + (\Gamma1) n\Gamma1 is the Euler characteristic of an (n \Gamma 1)­sphere.
Proof. By Theorem 4.5.1, H k (WK ) = H k (WK ) and H m+n\Gammak (WK ; @ c WK ) =
H m+n\Gammak (ZK ; T m ). Moreover, it can be seen in the same way as in Corollary 4.3.2
that relative Poincar'e duality isomorphisms (70) regard the bigraded structures in
the (co)homology of WK and (ZK ; T m ). It follows that
b \Gammaq;2p (WK ) = b \Gamma(m\Gamman)+q;2(m\Gammap) (ZK ; T m );
ü p (WK ) = (\Gamma1) m\Gamman üm\Gammap (ZK ; T m );
ü(WK ; t) = (\Gamma1) m\Gamman t 2m ü(ZK ; T m ; 1
t ):
(71)
Using (58) we calculate
(\Gamma1) m\Gamman t 2m ü(ZK ; T m ; 1
t )
= (\Gamma1) m\Gamman t 2m (1 \Gamma t \Gamma2 ) m\Gamman (h 0 + h 1 t \Gamma2 + \Delta \Delta \Delta + hn t \Gamma2n )
\Gamma (\Gamma1) m\Gamman t 2m (1 \Gamma t \Gamma2 ) m
= (1 \Gamma t 2 ) m\Gamman (h 0 t 2n + h 1 t 2n\Gamma2 + \Delta \Delta \Delta + hn ) + (\Gamma1) n\Gamma1 (1 \Gamma t 2 ) m :
Substituting the formula for ü(WK ; t) from Theorem 4.5.3 and the above expression
into (71) we obtain
(1 \Gamma t 2 ) m\Gamman (h 0 + h 1 t 2 + \Delta \Delta \Delta + hn t 2n ) +
\Gamma ü(K) \Gamma 1
\Delta (1 \Gamma t 2 ) m
= (1 \Gamma t 2 ) m\Gamman (h 0 t 2n + h 1 t 2n\Gamma2 + \Delta \Delta \Delta + hn ) + (\Gamma1) n\Gamma1 (1 \Gamma t 2 ) m :
Calculating the coefficient of t 2i in both sides after dividing the above identity by
(1 \Gamma t 2 ) m\Gamman , we obtain hn\Gammai \Gamma h i = (\Gamma1) i
\Gamma ü(K n\Gamma1 ) \Gamma ü(S n\Gamma1 )
\Delta \Gamma n
i
\Delta , as needed.
If jKj = S n\Gamma1 or n \Gamma 1 is odd, Corollary 4.5.4 gives the classical equations hn\Gammai = h i .
Corollary 4.5.5. If K n\Gamma1 is a simplicial manifold with h­vector (h 0 ; : : : ; hn ) then
hn\Gammai \Gamma h i = (\Gamma1) i (h n \Gamma 1)
\Gamma n
i
\Delta ; i = 0; 1; : : : ; n:
Proof. Since ü(K n\Gamma1 ) = 1 + (\Gamma1) n\Gamma1 hn , ü(S n\Gamma1 ) = 1 + (\Gamma1) n\Gamma1 , we have
ü(K n\Gamma1 ) \Gamma ü(S n\Gamma1 ) = (\Gamma1) n\Gamma1 (h n \Gamma 1) = (hn \Gamma 1)
(the coefficient (\Gamma1) n\Gamma1 can be dropped since for odd n \Gamma 1 the left hand side is
zero).
Corollary 4.5.6. For any (n \Gamma 1)­dimensional orientable simplicial manifold the
numbers hn\Gammai \Gamma h i , i = 0; 1; : : : ; n, are homotopy invariants. In particular, they do
not depend on a triangulation.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 77
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma \Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma \Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma \Gamma
\Gamma
\Gamma
\Gamma
Figure 10. Triangulation of T 2 with f = (9; 27; 18), h = (1; 6; 12; \Gamma1).
In the particular case of PL­manifolds the topological invariance of numbers
hn\Gammai \Gamma h i was firstly observed by Pachner in [67, (7.11)].
Example 4.5.7. Consider triangulations of the 2­torus T 2 . We have n = 3,
ü(T 2 ) = 0. From ü(K n\Gamma1 ) = 1 + (\Gamma1) n\Gamma1 hn we deduce h 3 = \Gamma1. Corollary 4.5.4
gives
h 3 \Gamma h 0 = \Gamma2; h 2 \Gamma h 1 = 6:
For instance, the triangulation on Figure 10 has f 0 = 9 vertices, f 1 = 27 edges and
f 2 = 18 triangles. The corresponding h­vector is (1; 6; 12; \Gamma1).
5. Subspace arrangements and cohomology rings of their
complements
5.1. Summary of results on the cohomology of general arrangement com­
plements. An arrangement is a finite set A = fL 1 ; : : : ; L r g of planes (affine
subspaces) in some affine space (either real or complex). For any arrangement
A = fL 1 ; : : : ; L r g in C m define its support jAj as
jAj :=
r
[
i=1
L i ae C m ;
and its complement U (A) as
U (A) := C m n jAj;
and similarly for arrangements in R m .
Arrangements and their complements play a pivotal r“ole in many constructions
of combinatorics, algebraic and symplectic geometry etc.; they also arise as config­
uration spaces for different classical mechanical systems. In the study of arrange­
ments it is very important to get a sufficiently 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 di­
rection appeared during the last three decades, however, the whole picture is far
from being complete. The theory ascends to work of Arnold [4], which described
the classifying space for the braid group Bn as the complement of arrangement of
all diagonal hyperplanes fz i = z j g in C n . The cohomology ring of this complement
was also calculated there. This result was generalised by Brieskorn [18] and moti­
vated the further development of the theory of complex hyperplane arrangements
(i.e. arrangements of codimension­one complex affine subspaces). One of the main
results here is the following.

78 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Theorem 5.1.1 ([4], [18], [64]). Let A = fL 1 ; : : : ; L r g be an arrangement of com­
plex hyperplanes in C m , and the hyperplane L i is the zero set of linear function l i ,
i = 1; : : : ; r. Then the integer cohomology algebra of the complement C m n jAj is
isomorphic to the algebra generated by closed differential 1­forms 1
2úi
dl i
l i .
Relations between the forms 1
2úi
dl i
l i
are also explicitly described. In the case of
diagonal hyperplanes fz i = z j g we have the forms ! ij = 1
2úi
d(z i \Gammaz j )
z i \Gammaz j
, which are
subject to the following relations:
! ij “ ! jk + ! jk “ ! ki + ! ki “ ! ij = 0:
The theory of complex hyperplane arrangements is probably the most well un­
derstood part of the whole study. Several surveys and monographs are available; we
mention just [65], where further references can be found. Relationships of real hy­
perplane arrangements with polytopes and oriented matroids are discussed in [89,
Lecture 7].
In the general situation, the Goresky--MacPherson theorem [42, Part III] ex­
presses the cohomology groups H i (U (A)) (without ring structure) as a sum of
homology groups of subcomplexes of a certain simplicial complex. We formulate
this result below. For a detailed survey of general arrangements we refer to [14].
Some important results in this direction can be found in monograph [87].
Let A = fL 1 ; : : : ; L r g be an arrangement of planes in R n . The intersections
v = L i 1
`` \Delta \Delta \Delta `` L i k
form a poset (P; !) with respect to the inclusion (i.e. v ! w if and only if v and
w are different and v is contained in w). The poset P is assumed to have a unique
maximal element T corresponding to the ambient space of the arrangement. The
rank function d on P is defined by d(v) = dimv. The order complex K(P) (see
Example 1.2.7) is called the order complex of arrangement A. Define
P (v;w) = fx 2 P : v ! x ! wg; P?v = fx 2 P : x ? vg:
Theorem 5.1.2 (Goresky and MacPherson [42, Part III]). The following formula
holds for the homology of the complement U (A):
H i
\Gamma
U (A); Z
\Delta
=
M
v2P
H n\Gammad(v)\Gammai\Gamma1
\Gamma
K(P?v ); K(P v;T ); Z
\Delta
;
with the agreement that H \Gamma1 (;; ;) = Z.
The proof of this theorem uses the stratified Morse theory developed in [42].
Remark. The homology groups of a complex arrangement in C n can be calculated
by regarding it as a real arrangement in R 2n .
The cohomology rings of the complements of arrangements are much more subtle.
In general, the integer cohomology ring of U (A) is not determined by the poset P.
An approach to calculating the cohomology algebra of the complement U (A) was
proposed by De Concini and Procesi [34]. In particular, they proved that the rational
cohomology ring of U (A) is determined by the combinatorics of intersections. This
result was extended by Yuzvinsky in [88].

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 79
5.2. Coordinate subspace arrangements and cohomology of ZK . An ar­
rangement A = fL 1 ; : : : ; L r g is called coordinate if every plane L i , i = 1; : : : ; r, is
a coordinate subspace. In this section we apply the results of chapter 4 to coho­
mology algebras of the complements of complex coordinate subspace arrangements.
The case of real coordinate arrangements is discussed at the end of the section.
Any coordinate subspace of C m has the form
L I = f(z 1 ; : : : ; z m ) 2 C m : z i 1
= \Delta \Delta \Delta = z i k = 0g;
(72)
where I = fi 1 ; : : : ; i k g is a subset of [m]. Obviously, dimL I = m \Gamma #I.
Construction 5.2.1. For each simplicial complex K on the set [m] define the
complex coordinate subspace arrangement CA(K) as the set of subspaces L I such
that I is not a simplex of K:
CA(K) = fL I : I =
2 Kg:
Denote the complement of CA(K) by U (K), that is
U (K) = C m n
[
I =
2K
L I :
(73)
If K 0 ae K is a subcomplex, then U (K 0 ) ae U (K).
Proposition 5.2.2. The assignment K 7! U (K) defines a one­to­one order­
preserving correspondence between simplicial complexes on the set [m] and com­
plements of coordinate subspace arrangements in C m .
Proof. Suppose CA is a coordinate subspace arrangement in C m . Define
K(CA) := fI ae [m] : L I 6ae jCAjg:
(74)
Obviously, K(CA) is a simplicial complex. By the construction, K(CA) depends
only on jCAj (i.e. on U (CA)) and U (K(CA)) = U (CA), whence the proposition
follows.
If a coordinate subspace arrangement A contains a hyperplane, say fz i = 0g,
then its complement U (A) is factorised as U (A 0 ) \Theta C \Lambda , where A 0 is a coordinate
subspace arrangement in the hyperplane fz i = 0g and C \Lambda = C n f0g. Thus, for any
coordinate subspace arrangement A the complement U (A) decomposes as
U (A) = U (A 0 ) \Theta (C \Lambda ) k ;
were A 0 is a coordinate arrangement in C m\Gammak that does not contain hyperplanes.
On the other hand, (74) shows that CA contains the hyperplane fz i = 0g if and
only if fig is not a vertex of K(CA). 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 ourselves to coordinate
subspace arrangements without hyperplanes and simplicial complexes on the vertex
set [m].
Remark. In terms of Construction 3.5.1 we have U (K) = K ffl (C ; C \Lambda ).
Example 5.2.3. 1. If K = \Delta m\Gamma1 (the (m \Gamma 1)­simplex) then U (K) = C m .
2. If K = @ \Delta m\Gamma1 (the boundary of simplex) then U (K) = C m n f0g.
3. If K is a disjoint union of m vertices, then U (K) is obtained by removing all
codimension­two coordinate subspaces z i = z j = 0, i; j = 1; : : : ; m, from C m .

80 VICTOR M. BUCHSTABER AND TARAS E. PANOV
e u
u u
­
¸¸¸¸¸¸¸ :
¸¸¸¸¸¸¸¸¸ ¸
\Phi \Phi \Phi \Phi \Phi \Phi \Phi*
\Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi
ae
ae ae
ae ae
ae ae?
ae
ae ae
ae ae
ae ae
ae ae ae
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma`
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma \Gamma
6
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambdaš
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda
\Lambda \Lambda
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta–
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta
\Delta \Delta
'
'
'
'
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'
'7
'
'
'
'
'
'
'
'
'
'
Figure 11. The retraction r : U (K) `` R m
+ ! cc(K) for K = @ \Delta m\Gamma1 .
The action of the algebraic torus (C \Lambda ) m on C m descends to U (K). In particular,
the standard action of the torus T m is defined on U (K). The quotient U (K)=T m
can be identified with U (K) `` R m
+ , where R m
+ is regarded as a subset of C m .
Lemma 5.2.4. cc(K) ae U (K) `` R m
+ and ZK ae U (K) (see Construction 1.5.9
and (39)).
Proof. Take y = (y 1 ; : : : ; ym ) 2 cc(K). Let I = fi 1 ; : : : ; i k g be the maximal subset
of [m] such that y 2 L I `` R n
+ (i.e. y i 1 = \Delta \Delta \Delta = y i k = 0). Then it follows from the
definition of cc(K) (see (22)) that I is a simplex of K. Hence, L I =
2 CA(K) and
y 2 U (K). Thus, the first statement is proved. Since cc(K) is the quotient of ZK ,
the second assertion follows from the first one.
Theorem 5.2.5. There is an equivariant deformation retraction U (K) ! ZK .
Proof. First, we construct a deformation retraction r : U (K) `` R m
+ ! cc(K). This
is done inductively. We start from the boundary complex of an (m \Gamma 1)­simplex
and remove simplices of positive dimensions until we obtain K. On each step we
construct a deformation retraction, and the composite map would be a required
retraction r.
If K = @ \Delta m\Gamma1 is the boundary complex of an (m\Gamma1)­simplex, then U (K)``R m
+ =
R m
+ n f0g. In this case the retraction r is shown on Figure 11. Now suppose that
K is obtained by removing one (k \Gamma 1)­dimensional simplex J = fj 1 ; : : : ; j k g from
simplicial complex K 0 , that is K [ J = K 0 . By the inductive hypothesis, the there
is a deformation retraction r 0 : U (K 0 ) `` R m
+ ! cc(K 0 ). Let a 2 R m
+ be the point
with coordinates y j1 = : : : = y jk = 0 and y i = 1 for i =
2 J . Since J is not a simplex
of K, we have a =
2 U (K) `` R m
+ . At the same time, a 2 C J (see (19)). Hence, we may
apply the retraction from Figure 11 on the face C J ae I m , with centre at a. Denote
this retraction by r J . Then r = r J ffi r 0 is the required deformation retraction.
The deformation retraction r : U (K) `` R m
+ ! cc(K) is covered by an equivariant
deformation retraction U (K) ! ZK .
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 5.2.5
can be realised as the orbit map for an action of a contractible group. We denote
U (P n ) := U (KP ). Set
R m
? = f(y 1 ; : : : ; y m ) 2 R m : y i ? 0; i = 1; : : : ; mg ae R m
+ :

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 81
Then R m
? is a group with respect to the multiplication, and it acts on R m , C m and
U (P n ) by coordinatewise multiplication. There is the isomorphism exp : R m ! R m
?
between the additive and the multiplicative groups taking (y 1 ; : : : ; y m ) 2 R m to
(e y1 ; : : : ; e ym ) 2 R m
? .
Let us consider the m \Theta (m \Gamma n)­matrix W introduced in Construction 1.1.4 for
any simple polytope (1).
Proposition 5.2.6. For any vertex v = F i 1
`` \Delta \Delta \Delta `` F i n of P n the maximal minor
W “ i 1 ::: “ i n which is obtained by deleting n rows i 1 ; : : : ; i n from W is non­degenerate:
det W “ i 1 ::: “ i n
6= 0.
Proof. If det W “ i 1 ::: “ i n = 0 then vectors l i 1 ; : : : ; l i n (see (1)) are linearly dependent,
which is impossible.
The matrix W defines the subgroup
RW =
\Phi
(e w11 Ü1 +\Delta\Delta\Delta+w 1;m\Gamman Üm\Gamman ; : : : ; e wm1 Ü1 +\Delta\Delta\Delta+w m;m\Gamman Üm\Gamman )
\Psi ae R m
? ;
(75)
where the parameters Ü 1 ; : : : ; Ü m\Gamman vary over R m\Gamman . Obviously, RW ¸ = R m\Gamman
? .
Theorem 5.2.7 ([22, Theorem 2.3]). The subgroup RW ae R m
? acts freely on
U (P n ) ae C m . The composite map ZP ,! U (P n ) ! U (P n )=RW of the embed­
ding i e (Lemma 3.1.6) and the orbit map is an equivariant diffeomorphism (with
respect to the T m ­actions).
Proof. A point from C m has the non­trivial isotropy subgroup with respect to
the action of R m
? on C m if and only if at least one of its coordinates vanishes.
It follows from (73) that if a point x 2 U (P n ) has some zero coordinates, then
the corresponding facets of P n have at least one common vertex v 2 P n . Let
v = F i 1
`` \Delta \Delta \Delta `` F i n . The isotropy subgroup of the point x with respect to the
action of the subgroup RW is non­trivial only if some linear combination of columns
of W lies in the coordinate subspace spanned by e i 1 ; : : : ; e i n . But this implies
that det W “ i 1 ::: “ i n = 0, which contradicts Proposition 5.2.6. Thus, RW acts freely on
U (P n ).
To prove the second part of the theorem it is sufficient to show that each orbit
of the action of RW on U (P n ) ae C m intersects the image i e (ZP ) at a single point.
Since the embedding i e is equivariant with respect to the T m ­actions, the latter
statement is equivalent to that each orbit of the action of RW on U (P n )``R m
+ inter­
sects the image i P (P n ) (see Theorem 1.5.6) at a single point. Let y 2 i P (P n ) ae R m .
Then y = (y 1 ; : : : ; ym ) lies in some n­face i P (C n
v ) of the cube I m ae R m , see (20).
We need to show that the (m \Gamma n)­dimensional subspace spanned by the vectors
(w 11 y 1 ; : : : ; wm1 ym ) t ; : : : ; (w 1;m\Gamman y 1 ; : : : ; wm;m\Gamman ym ) t is in general position with
the n­face i P (C n
v ) of I m . This follows directly from (20) and Proposition 5.2.6.
Suppose now that P n is a lattice simple polytope, and let MP be the correspond­
ing toric variety (Construction 2.1.3). Along with the real subgroup RW ae R m
? (75)
we define
CW =
\Phi (e w11 OE 1 +\Delta\Delta\Delta+w 1;m\Gamman OE m\Gamman ; : : : ; e wm1 OE 1 +\Delta\Delta\Delta+w m;m\Gamman OE m\Gamman )
\Psi ae (C \Lambda ) m ;
where the parameters OE 1 ; : : : ; OE m\Gamman vary over C m\Gamman . Obviously, CW ¸ = (C \Lambda ) m\Gamman .
It is shown in [7], [11], [31] that CW acts freely on U (P n ) and the toric variety MP

82 VICTOR M. BUCHSTABER AND TARAS E. PANOV
can be identified with the orbit space (or geometric quotient) U (P n )=CW . Thus,
we have the following commutative diagram:
U (P n ) RW ¸ =R m\Gamman
?
\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma! ZP
CW ¸ =(C \Lambda ) m\Gamman
? ? y
? ? yT m\Gamman
MP MP :
(76)
Remark. It can be shown [31, Theorem 2.1] that any toric variety M \Sigma correspond­
ing to a fan \Sigma ae R n with m one­dimensional cones can be identified with the
universal categorical quotient U (CA \Sigma )=G, where U (CA \Sigma ) is the complement of a
certain coordinate arrangement (determined by the fan \Sigma) and G ¸ = (C \Lambda ) m\Gamman . The
categorical quotient becomes the geometric quotient if and only if the fan \Sigma is
simplicial. In this case U (CA \Sigma ) = U (K \Sigma ).
On the other hand, if the projective toric variety MP is non­singular, then MP is
a symplectic manifold of dimension 2n, and the action of T n on it is Hamiltonian [7].
In this case the diagram (76) displays MP as the result of the process of symplectic
reduction. Namely, let HW ¸ = T m\Gamman be the maximal compact subgroup in CW , and
¯ : C m ! R m\Gamman the moment map for the Hamiltonian action of HW on C m . Then
for any regular value a 2 R m\Gamman of the map ¯ there is the following diffeomorphism:
¯ \Gamma1 (a)=HW \Gamma! U (P n )=CW = MP
(details can be found in [7]). In this situation ¯ \Gamma1 (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 realised as the quotient of a non­singular
projective toric variety).
Example 5.2.8. Let P n = \Delta n (the n­simplex). Then m = n+ 1, U (P n ) = C n+1 n
f0g. RW ¸ = R? , CW ¸ = C \Lambda and HW ¸ = S 1 are the diagonal subgroups in R n+1
? ,
(C \Lambda ) n+1 and T m+1 respectively (see Example 1.1.5). Hence, ZP ¸ = S 2n+1 = (C n+1 n
f0g)=R ? and MP = (C n+1 n f0g)=C \Lambda = CP n . The moment map ¯ : C m ! R takes
(z 1 ; : : : ; z m ) 2 C m to 1
2 (jz 1 j 2 +: : :+jz m j 2 ), and for a 6= 0 we have ¯ \Gamma1 (a) ¸ = S 2n+1 ¸ =
ZK .
The previous discussion illustrates the importance of calculating the cohomology
of subspace arrangement complements.
Theorem 5.2.9 (Buchstaber and Panov). The following isomorphism of graded
algebras holds:
H \Lambda
\Gamma U (K)
\Delta ¸ = Tor k[v1 ;::: ;vm ]
\Gamma k(K); k
\Delta
= H
\Theta \Lambda[u 1 ; : : : ; um
]\Omega k(K); d
\Lambda :
Proof. This follows from theorems 5.2.5, 4.2.1 and 4.2.2.
Theorem 5.2.9 provides an extremely effective way to calculate the cohomology
algebra of the complement of any complex coordinate subspace arrangement. The
De Concini and Procesi [34] and Yuzvinsky [88] rational models of the cohomology
algebra of an arrangement complement also can be interpreted as an application of
the Koszul resolution. However, these author did not discuss the relationships with
the Stanley--Reisner ring in the case of coordinate subspace arrangements.

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 83
Problem 5.2.10. Calculate the cohomology algebra with Zcoefficients of a co­
ordinate subspace arrangement complement and describe its relationships with the
corresponding Tor­algebra Tor Z[v1 ;::: ;vm ] (Z(K); Z).
Example 5.2.11. Let K be a disjoint union of m vertices. Then U (K) is obtained
by removing all codimension­two coordinate subspaces z i = z j = 0, i; j = 1; : : : ; m
from C m (see Example 5.2.3). The face ring is k(K) = k[v 1 ; : : : ; v m ]=I K , where
IK is generated by monomials v i v j , i 6= j. An easy calculation using Corol­
lary 5.2.9 shows that the subspace of cocycles in
k(K)\Omega \Lambda[u 1 ; : : : ; um ] has the
basis consisting of monomials v i 1 u i 2 u i 3 \Delta \Delta \Delta u i k with k – 2, i p 6= i q for p 6= q. Since
deg(v i 1 u i 2 u i 3
\Delta \Delta \Delta u i k ) = k + 1, the space of (k + 1)­dimensional cocycles has dimen­
sion m
\Gamma m\Gamma1
k\Gamma1
\Delta . The space of (k + 1)­dimensional coboundaries is
\Gamma m
k
\Delta ­dimensional (it
is spanned by the coboundaries of the form d(u i 1 \Delta \Delta \Delta u i k )). Hence,
dimH 0
\Gamma
U (K)
\Delta
= 1; H 1
\Gamma
U (K)
\Delta
= H 2
\Gamma
U (K)
\Delta
= 0;
dimH k+1
\Gamma
U (K)
\Delta
= m
\Gamma m\Gamma1
k\Gamma1
\Delta \Gamma
\Gamma m
k
\Delta
= (k \Gamma 1)
\Gamma m
k
\Delta
; 2 Ÿ k Ÿ m;
and the multiplication in the cohomology is trivial.
In particular, for m = 3 we have 6 three­dimensional cohomology classes [v i u j ],
i 6= j subject to 3 relations [v i u j ] = [v j u i ], and 3 four­dimensional cohomology
classes [v 1 u 2 u 3 ], [v 2 u 1 u 3 ], [v 3 u 1 u 2 ] subject to one relation
[v 1 u 2 u 3 ] \Gamma [v 2 u 1 u 3 ] + [v 3 u 1 u 2 ] = 0:
Hence, dimH 3 (U (K)) = 3, dimH 4 (U (K)) = 2, and the multiplication is trivial.
Example 5.2.12. Let K be the boundary of an m­gon, m ? 3. Then
U (K) = C m n
[
i\Gammaj 6=0;1 mod m
fz i = z j = 0g:
By Theorem 5.2.9, the cohomology ring of H \Lambda (U (K); k) is isomorphic to the ring
described in Example 4.3.5.
As it is shown in [41], in the case of arrangements of real coordinate subspaces
only additive analogue of our Theorem 5.2.9 holds. Namely, let us consider the
polynomial ring k[x 1 ; : : : ; xm ] with deg x i = 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) are calculated by means of the following
result.
Theorem 5.2.13 ([41, Theorem 3.1]). The following isomorphism hold:
H p
\Gamma
UR (K)
\Delta ¸ =
X
\Gammai+j=p
Tor \Gammai;j
k[x1 ;::: ;xm ]
\Gamma k(K); k
\Delta = H \Gammai;j
\Theta \Lambda[u 1 ; : : : ; um
]\Omega k(K); d
\Lambda
;
where bideg u i = (\Gamma1; 1), bideg v i = (0; 1), du i = x i , dx i = 0.
As it was observed in [41], there is no multiplicative isomorphism analogous to
Theorem 5.2.9 in the case of real arrangements, that is, the algebras H \Lambda (UR (K)) and
Tor k[x1 ;::: ;xm ] (k(K); k) are not isomorphic in general. The paper [41] also contains
the formulation of the first multiplicative isomorphism of our Theorem 5.2.9 for
complex coordinate subspace arrangements (see [41, Theorem 3.6]), with a reference
to yet unpublished paper by Babson and Chan.
Until now, we have used the description of coordinate subspaces by means of
equations (see (72)). On the other hand, a coordinate subspace can be defined as

84 VICTOR M. BUCHSTABER AND TARAS E. PANOV
the linear span of some subset of the standard basis fe 1 ; : : : ; e m g. This leads to
the dual approach to the description of coordinate subspace arrangements, which
corresponds to the passage from simplicial complex K to the dual complex b
K (Ex­
ample 1.2.4). This approach was used in [35]. It was shown there that the summands
in the Goresky--MacPherson formula in the coordinate subspace arrangement case
are homology groups of links of simplices of b
K. This allowed to interpret the prod­
uct of cohomology classes of the complement of a coordinate subspace arrangement
(either real or complex) in terms of the combinatorics of links of simplices in b
K
(see [35, Theorem 1.1]).
We mention that our theorems 4.2.2 and 5.2.5 show that the Goresky--Mac­
Pherson result (Theorem 5.1.2) in the case of coordinate subspace arrangements is
equivalent to the Hochster theorem (Theorem 1.4.5).
5.3. Diagonal subspace arrangements and cohomology of loop
space\Omega ZK .
In this section we establish relationships between the results of [70] on the coho­
mology of real diagonal arrangement complements and the cohomology of the loop
spaces\Omega B T ZK
and\Omega ZK .
For each subset I = fi 1 ; : : : ; i k g ae [m] define the diagonal subspace D I in R m as
D I = f(y 1 ; : : : ; ym ) 2 R m : y i 1 = \Delta \Delta \Delta = y i k g:
Diagonal subspaces in C m are defined similarly. An arrangement of planes A =
fL 1 ; : : : ; L r g (either real or complex) is called diagonal if all planes L i , i = 1; : : : ; r,
are diagonal subspaces. The classical example of a diagonal subspace arrangement
is given by the arrangement of all diagonal hyperplanes fz i = z j g in C m ; its com­
plement is the classifying space for the braid group Bm , see [4].
Construction 5.3.1. Given a simplicial complex K on the vertex set [m], intro­
duce the diagonal subspace arrangement DA(K) as the set of subspaces D I such
that I is not a simplex of K:
DA(K) = fD I : I =
2 Kg:
Denote the complement of the arrangement DA(K) by M (K).
The proof of the following statement is similar to the proof of the corresponding
statement for coordinate subspace arrangements from section 5.2.
Proposition 5.3.2. The assignment K 7! M (K) defines a one­to­one order­
preserving correspondence between simplicial complexes on the vertex set [m] and
the complements of diagonal subspace arrangements in R m .
Here we still assume that k is a field. The multigraded (or N m ­graded) struc­
ture in the ring k[v 1 ; : : : ; vm ] (Construction 1.4.7) defines an N m ­grading in the
Stanley--Reisner ring k(K). The monomial v i 1
1 \Delta \Delta \Delta v i m
m acquires the multidegree
(2i 1 ; : : : ; 2i m ). Let us consider the modules Tor k(K) (k; k). They can be calculated,
for example, by means of the minimal free resolution (Example 1.4.2) of the field k
regarded as a k(K)­module. The minimal resolution also carries a natural N m ­
grading, and we denote the subgroup of elements of multidegree (2i 1 ; : : : ; 2i m ) in
Tor k(K) (k; k) by Tor k(K) (k; k) (2i1 ;::: ;2i m ) .
Theorem 5.3.3 ([70, Theorem 1.3]). The following isomorphism holds for the co­
homology groups of the complement M (K) of a real diagonal subspace arrangement:
H i \Gamma M (K); k
\Delta ¸ = Tor \Gamma(m\Gammai)
k(K)
(k; k) (2;::: ;2) :

TORUS ACTIONS, COMBINATORIAL TOPOLOGY AND HOMOLOGICAL ALGEBRA 85
Remark. Instead of simplicial complexes K on the vertex set [m] the authors of [70]
considered square­free monomial ideals I ae k[v 1 ; : : : ; v m ]. Proposition 1.3.2 shows
that these two approaches are equivalent.
Theorem 5.3.4. The following additive isomorphism holds:
H
\Lambda(\Omega BT ZK ; k) ¸ = Tor k(K) (k; k):
Proof. Let us consider the Eilenberg--Moore spectral sequence of the Serre fibra­
tion P ! SR(K) with
fibre\Omega SR(K), where SR(K) is the Stanley--Reisner space
(Definition 3.4.2) and P is the path space over SR(K). By Corollary 4.1.4,
E 2 = Tor H \Lambda (SR(K))
\Gamma H \Lambda (P ); k
\Delta ¸ = Tor k(K) (k; k);
(77)
and the spectral sequence converges to Tor C \Lambda (SR(K)) (C \Lambda (P ); k) ¸ = H
\Lambda(\Omega SR(K)).
Since P is contractible, there is a cochain equivalence C \Lambda (P ) ' k. We have
C \Lambda (SR(K)) ¸ = k(K). Therefore,
Tor C \Lambda (SR(K))
\Gamma C \Lambda (P ); k
\Delta ¸ = Tor k(K) (k; k);
which together with (77) shows that the spectral sequence collapses ate the E 2
term. Hence, H
\Lambda(\Omega SR(K)) ¸ = Tor k(K) (k; k). Finally, Theorem 3.4.4 shows that
H
\Lambda(\Omega SR(K)) ¸ = H
\Lambda(\Omega B T ZK ), which concludes the proof.
Proposition 5.3.5. The following isomorphism of algebras holds
H
\Lambda(\Omega B T ZK ) ¸ = H
\Lambda(\Omega ZK
)\Omega \Lambda[u 1 ; : : : ; um ]:
Proof. Consider the bundle B T ZK ! BT m with fibre ZK . It is not hard to prove
that the corresponding loop
bundle\Omega B T ZK ! T m with
fibre\Omega ZK is trivial (note
that\Omega BT m ¸ = T m ). To finish the proof it remains to mention that H \Lambda (T m ) ¸ =
\Lambda[u 1 ; : : : ; um ].
Theorems 5.2.5 and 5.2.9 give an application of the theory of moment­angle
complexes to calculating the cohomology ring of a coordinate subspace arrange­
ment complement. Similarly, theorems 5.3.3, 5.3.4 and Proposition 5.3.5 establish
the connection between the cohomology of a diagonal subspace arrangement com­
plement and the cohomology of the loop space over the moment­angle complex ZK .
However, in this case the situation is much more subtle than in the case of coordinate
subspace arrangements. For instance, we do not have an analogue of the multiplica­
tive isomorphism from Theorem 5.2.9. That is why we consider this concluding
section only as the first step in applying the theory of moment­angle complexes to
studying the complements of diagonal subspace arrangements.
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Department of Mathematics and Mechanics, Moscow State University, 119899
Moscow RUSSIA
E­mail address: buchstab@mech.math.msu.su
E­mail address: tpanov@mech.math.msu.su