Документ взят из кэша поисковой машины. Адрес оригинального документа : http://higeom.math.msu.su/people/taras/mypapers/6sppo.pdf
Дата изменения: Thu Oct 6 06:48:42 2005
Дата индексирования: Sun Apr 10 00:55:35 2016
Кодировка:

SPACES OF POLYTOPES AND COBORDISM OF TORIC MANIFOLDS
VICTOR M BUCHSTABER, TARAS E PANOV, AND NIGEL RAY Abstract. Our aim is to bring the theory of analogous polytopes to bear on the study of omnioriented toric manifolds. By way of application, we simplify and correct certain proofs of the first and third author on the representability of complex cobordism classes. These proofs concern quotient polytopes; the first involves framed embeddings in the positive cone, and the second considers the role of orientations in forming connected sums. Analogous polytopes provide an illuminating context within which to deal with several of the details. Our modified connected sum incorporates an oriented cube of appropriate dimension, and is directly relevant to the proof that any complex cobordism class may be represented by an omnioriented toric manifold. We illustrate the results by means of 4dimensional examples.

1. Introduction The theory of analogous polytopes was initiated by Alexandrov [1] in the 1930s, and extended more recently by Khovanskii and Puhlikov [10]. Our aim is to apply this theory to the algebraic topology of torus actions, in the context of Davis and Januszkiewicz's work [5] on toric geometry. Davis and Januszkiewicz explain how to construct a 2ndimensional toric manifold M from a characteristic pair (P, ), where P is a simple convex polytope of dimension n, and is a function with certain special properties which assigns a subcircle of the torus T n to each facet of P . By construction M admits a locally standard T n action, whose quotient space is homeomorphic to P . Every such polytope is equivalent to an arrangement H of m closed halfspaces in Rn , whose bounding hyperplanes meet only in general position. The intersection of the half-spaces is assumed to be bounded, and defines P . The (n - 1) dimensional faces form the facets of P , and general position ensures that any face of codimension k is the intersection of precisely k facets. In particular, every vertex is the intersection of n facets, and lies in an open neighbourhood isomorphic to the positive cone Rn . For any characteristic pair (P, ), it is possible to vary P within its combinatorial equivalence class without affecting the -equivariant diffeomorphism type [5] of the toric manifold M . For a fixed arrangement, we consider the vector dH of signed distances from the origin O to the bounding hyperplanes in Rn ; a coordinate is positive when
Key words and phrases. analogous polytopes, complex cobordism, connected sum, framing, omniorientation, stable tangent bundle, toric manifold. The first and second authors were supported by the Russian Foundation for Basic Research, grants number 04-01-00702 and 05-01-01032. The second author was supported by an EPSRC Visiting Fellowship at the University of Manchester.
1

2

VICTOR M BUCHSTABER, TARAS E PANOV, AND NIGEL RAY

O lies in the interior of the corresponding half-space, and negative in the complement. We then identify the m-dimensional vector space RH with the space of arrangements analogous to H. Under this identification, dH corresponds to H itself, and every other vector corresponds to the arrangement obtained by the appropriate parallel displacement of half-spaces. For small displacements, the intersections of the half-spaces are polytopes similar to P . For larger displacements the intersections may be degenerate, or empty; in either case, they are known as virtual polytopes, analogous to P . In [4], the first and third authors consider dicharacteristic pairs (P, ), where is replaced by a homomorphism : T H T n . This has the effect of orienting each of the subcircles (Fj ) of T n , and leads to the construction of an omnioriented toric manifold M ; [4, Theorem 3.8] claims that a canonical stably complex structure may then be chosen for M . The proof, however, has two flaws. Firstly, it fails to provide a sufficiently detailed explanation of how a certain complexified neighbourhood of P may be framed, and secondly, it requires an orientation of M (and hence of P ) for the stably complex stucture to be uniquely defined. The latter issue has already been raised in [2, §5.3], but amended proofs have not been given. One of our aims is to show that analogous polytopes offer a natural setting for some of the details. The main application of [4, Theorem 3.8] is as follows. Theorem 6.11 [4]. In dimensions > 2, every complex cobordism class contains a toric manifold, necessarily connected, whose stably complex structure is induced by an omniorientation, and is therefore compatible with the action of the torus. By [3], this result already holds for additive generators of the complex coborU dism groups n . So the proof proceeds by considering 2ndimensional omnioriented toric manifolds M1 and M2 , with quotient polytopes P1 and P2 respectively, and constructs a third such manifold M , which is complex cobordant to the connected sum M1 # M2 . For the quotient polytope of M , the authors use the connected sum P1 # P2 , over which the dicharacteristics naturally extend. In the light of the preceding observations, we must amend the proof so as to incorporate orientations of P1 and P2 . However, it is not always possible to form P1 # P2 in the oriented sense, and simultaneously extend the dicharacteristics. Instead, we replace M2 with a complex cobordant toric manifold M2 , whose quotient polytope is I n # P2 , where I n denotes an appropriately oriented n cube. It turns out that the resulting gain in geometrical freedom allows us to extend both orientations and dicharacteristics; the result is the omnioriented toric manifold M1 # M2 over the polytope P1 P2 = P1 # I n # P2 , which we call the box sum of P1 and P2 . We may then complete the proof of Theorem 6.11 as described in Section 5 below. In dimension 2, P1 P2 is combinatorially equivalent to the Minkowski sum P1 + P2 , which is central to the theory of analogous polytopes. In [4], the authors compare Theorem 6.11 with a famous question of Hirzebruch, who asks for a description of those complex cobordism classes which may be represented by connected algebraic varieties. This is a difficult problem, and

SPACES OF POLYTOPES AND COBORDISM

3

remains unsolved; nevertheless, our modification to the proof of Theorem 6.11 adds some value to the comparison, in the following sense. Given complex cobordism classes [N1 ] and [N2 ] of the same dimension, suppose that N1 and N2 are connected. Then we may form the connected sum N1 # N2 in the standard fashion, so that it is also a connected, stably complex manifold, and represents [N1 ] + [N2 ]. If, on the other hand, N1 and N2 are algebraic varieties, then N1 # N2 is not usually algebraic. In these circumstances we might proceed by analogy with the toric case, and look for an alternative representative N2 such that N1 # N2 is also algebraic. For the reader's convenience we retain most of the notation and conventions of [4], with two obvious exceptions concerning the half-spaces (2.1). Firstly, we assume that the set of half-spaces defining any polytope is ordered, and discuss the effects of choosing alternative orderings as required. Secondly, we reverse the direction of the inequalitites, so that the normals to the bounding hyperplanes point outwards, in agreement with Khovanskii. The contents of our sections are as follows. In section 2 we recall various definitions and notation concerning simple convex polytopes with ordered facets. We also introduce the space R(P ) of polytopes analogous to a fixed example P , and re-interpret the cokernel of an associated transformation. In section 3 we summarise Davis and Januszkiewicz's construction of a toric manifold M over a polytope P having m facets. We also offer a quadratic description of the auxiliary T m -space ZP , and define M as its quotient by the kernel of a dicharacteristic homomorphism. In Section 4 we amend the definition of omniorientation so as to include an orientation of M , and recall the stably complex structure which results. In so doing, we utilise a framing of P as a submanifold with corners of the positive cone in R(P ). We review the construction of connected sum for omnioriented toric manifolds in Section 5, by encoding the additional orientations as signs attached to the fixed points. By way of application, we correct the proof of [4, Theorem 3.8]. Finally, in Section 6, we exemplify the realisation of 4dimensional complex cobordism classes by omnioriented toric manifolds, and comment on analogous situations in higher dimensions. We are pleased to acknowledge the assistance of several colleagues in preparing this article. In particular, Tony Bahri, Kostya Feldman and Neil Strickland offered trenchant criticisms of [4], which alerted its authors to the need for clarification and correction. Peter McMullen also provided important guidance to offset the third author's lack of experience with simple polytopes. 2. Analogous polytopes We work in a real vector space V of dimension n, equipped with a euclidean inner product , . An arrangement H of closed half-spaces in V is a collection of subsets (2.1) Hi = {x V : ai , x bi } for 1 i m, where ai lies in V and bi is a real scalar. Unless stated otherwise, we shall assume that H has cardinality m n, and that ai has unit length for every

4

VICTOR M BUCHSTABER, TARAS E PANOV, AND NIGEL RAY

1 i m. We consider Hi as a smooth manifold, whose boundary Hi is its bounding hyperplane (2.2) Yi = {x V : ai , x = b} for 1 i m,

with outward pointing normal vector ai . If the intersection i Hi is bounded, it forms a convex polytope P ; otherwise, it is a polyhedron. We assume that P has maximal dimension n, and that none of the half-spaces is redundant, in the sense that no Hi may be deleted without enlarging P . In these circumstances, H and P are interchangeable. We may also specify P by a matrix inequality AP x bP , where AP is the m в n matrix of row vectors ai , and bP is the column vector of scalars bi in Rm . The positive cone Rn is an important polyhedron in Rn . It is determined by the half-spaces {x Rn : -ej , xj 0} for 1 j n,

and consists of the vectors {x : xj 0, 1 j n}. A supporting hyperplane is characterised by the property that P lies within one of its two associated half-spaces. A proper face of P is defined by its intersection with any supporting hyperplane, and forms a convex polytope of lower dimension. We regard P as an ndimensional face of itself; the faces of dimension 0, 1, and n - 1 are known as vertices, edges, and facets respectively. There is one facet Fi = P Yi for every bounding hyperplane (2.2), so the facets corresponds bijectively to the half-spaces (2.1). We deem a vertex v and facet Fi to be opposite whenever v lies in the interior of Hi . If the bounding hyperplanes are in general position, then every vertex of P is the intersection of exactly n facets, and has m - n opposite half-spaces. In these circumstances, P is simple. From this point on, we deal only with simple polytopes, and reserve the notation q = q (P ) and m = m(P ) for the number of vertices and the number of facets respectively. We assume that the ordering o of the half-spaces Hi is such that the intersection F1 · · · Fn of the first n facets is a vertex of P , and describe P as strongly ordered by o. We call v the initial vertex of P . The faces of codimension k may then be labelled with their list of defining facets and ordered lexicographically, for every 1 k n. In particular, the vertices of P are ordered by this procedure. With respect to inclusion, the faces form a poset LF (P ), with unique maximal element P . This poset fails to be a lattice only because we usually omit the empty face, which would otherwise form a unique minimal element. Two polytopes are combinatorial ly equivalent whenever their face posets are isomorphic; this occurs precisely when the polytopes are diffeomorphic as smooth ndimensional manifolds with corners. A combinatorial equivalence class of polytopes is known as a combinatorial polytope, and most of our constructions are defined on such classes. Nevertheless, it is usually helpful to keep a representative polytope in mind, rather than the underlying poset. Examples include the vertex figures Pv , which are formed by intersecting P with any closed halfspace whose interior contains a single vertex v . Because P is simple, Pv is an nsimplex for any v .

SPACES OF POLYTOPES AND COBORDISM

5

For computational purposes, it is sometimes convenient to locate the initial vertex of P at the origin, and use the first n normal vectors a1 , . . . , an as an orthonormal basis for V . This may be achieved by an appropriate affine transformation, and does not affect the combinatorial equivalence class of P . Fixing H, we consider the vector dH Rm , whose ith coordinate is the signed distance from the origin O to Yi in V , for 1 i m. The sign is positive when O lies in the interior of Hi , and negative in the exterior. So long as we maintain our convention that the normal vectors ai have unit length, dH coincides with bP ; otherwise, the distances have to be scaled accordingly. Every vector dH + h in Rm may then be identified with an analogous arrangement of half-spaces, defined by translating each of the half-spaces Hi by hi , for 1 i m. Some such arrangements determine convex polytopes P (h), and others, dubbed virtual polytopes, do not. In either case, they are described as being analogous to P . We note that P (h) is given by (2.3) {x V : AP x b P + h },

and is combinatorially equivalent to P when h is small, because P is simple. In particular, we have that P (0) = P . Examples 2.4. (1) The zero vector 0 Rm is identified with the central arrangement H0 , whose bounding hyperplanes contain the origin; the corresponding polytope P (-bP ) = {0} is virtual. The basis vector ei Rm is identified with the arrangement obtained from H0 by translating Hi by 1; the corresponding polytope P (-bP + ei ) = Pi may be virtual, or a simplex. (2) Any x V defines a vector AP x RH . Then bP - AP x is identified with the arrangement given by translating H by -x; the corresponding polytope P (-AP x) is the translate P - x, and is congruent to P . As x varies, we obtain an nparameter family of analogous polytopes, each of which is congruent to P. The Minkowski sum of subsets P , Q V is given by P + Q = {x + y : x P, y Q} V . If P and Q are convex polytopes, so is P + Q; moreover, when P is analogous to Q, so is P + Q. Under the identification of bP + h with P (h), vector addition corresponds to Minkowski sum, and scalar multiplication to rescaling. In this context, we denote the m-dimensional vector space of polytopes analogous to P by R(P ), and consider the identification as an isomorphism (2.5) k: R
m

- R(P ),

where

k (bP + h) = P (h).

We may interpret the matrix AP as a linear transformation V Rm . Since the points of P are specified by the constraint AP x bP , the intersection of the affine subspace bP - AP (V ) with the positive cone Rm is a copy of P in Rm . In other words, the formula iP (x) = bP - AP x defines an affine injection (2.6) iP : V - Rm ,

6

VICTOR M BUCHSTABER, TARAS E PANOV, AND NIGEL RAY

which embeds P as a submanifold with corners of the positive cone. Since iP maps the half-space Hi to the half-space {y : yi 0}, this embedding respects the codimension of faces. The composition P = k iP restricts to an affine injection P R(P ), and Example (2.4)(2) identifies P (x) as the polytope congruent to P , obtained by translating the origin to x, for all x in P . Of course, P (P ) is a submanifold with corners of the positive cone R(P ) . Given any shift vector h in Rm , the half-spaces Hi + hi are also ordered by o, and determine the initial vertex v (h) of P (h). For every 1 i m, we write di (h) for the signed distance between v (h) and the supporting hyperplane Yi + hi ; in other words, di (h) = bi + hi - ai , v (h) for all 1 i m, and d1 (h) = · · · = dn (h) = 0 by construction. We then define the linear transformation Co : Rm Rm-n by the formula (2.7) Co (bP + h) = (d
m-n+1

(h), . . . , dm (h)).

Using (2.5), we may interpret Co as a transformation R(P ) Rm-n , which acts by P (h) (dm-n+1 (h), . . . , dm (h)). Clearly, Co is epimorphic. Prop osition 2.8. As a transformation V R zero.
m-n

, the composition Co · AP is

Proof. The di (h) are metric invariants of the polytope P (h), so Co takes identical values on congruent polytopes. In particular, it is constant on the translates P - x for all values x V , and therefore on the affine plane bP - AP (V ). So Co (AP (V )) = 0, as required. Proposition 2.8 shows that o determines a short exact sequence
P 0 - V - R

A

mC

o - R

m-n

- 0,

or equivalently, a choice of basis for coker AP . A matrix (ci,j ) for Co is most easily computed by assuming that the normal vectors a1 , . . . , an form an orthonormal basis for V , as described in Section 2. Then the basis polytopes Pj of (2.4)(1) satisfy di (Pj ) = for all n + 1 i -ai,j i,j if 1 j if n + 1 n j m, .

(2.9)

m, and we deduce that -an+1,1 . . . -an+1,n 1 0 . . . 0 -an+2,1 . . . -an+2,n 0 1 . . . 0 (ci,j ) = . . . . .. . .. . . .. . .. . .. . . -am,1 . . . -am,n 0 0 . . . 1

An alternative strong ordering o provides an alternative basis for coker AP , and the corresponding matrix (ci,j ) is obtained by substituting an+1 , . . . , am into (2.9). This procedure sets up a bijection between strong orderings on P and matrices of the form (2.9).

SPACES OF POLYTOPES AND COBORDISM

7

Of course, any (m - n) в m matrix C of full rank for which C AP = 0 also yields a basis for coker AP , and satisfies the following property. Lemma 2.10. Let C be the (m-n)в(m-k ) matrix obtained from C by deleting columns cj1 , . . . , cjk , for some 1 k n; if the intersection Fj1 · · · Fjk is a face of P of codimension k , then C has rank m - n. Proof. Let : R
m-k

R

m

be the inclusion of the subspace {x : xj1 = · · · = xjk = 0}

and : Rm Rk composition C · , the rows aj1 , . . . , that · AP is an has rank m - n.

the associated quotient map. Then C is the matrix of the and the k в n matrix A of the composition · AP consists of ajk of AP . The data implies that A has rank k , and therefore epimorphism; so C · is also an epimorphism, and its matrix

3. Toric manifolds In this section we include a summary of Davis and Januszkiewicz's construction of toric manifolds over a simple polytope P . We appeal to their auxiliary space ZP , for which we provide an alternative description in terms of quadratic hypersurfaces. Throughout, we use the methods and notation of [4], under the additional assumption that P is strongly ordered by o. In particular, we denote the ith coordinate subcircle of the standard mtorus T m by Ti , for every 1 i m. For each point p in P , we define the subgroup T (p) by Ti < T m .
Hi p

If p is a vertex, then T (p) has maximal dimension n; if p is an interior point of P , then T (p) consists of the trivial subgroup {1}. Davis and Januszkiewicz introduce the identification space ZP as (3.1) T
m

в P / ,

where (t1 , p) (t2 , p) if and only if t-1 t2 T (p). So ZP is an (m + n)1 dimensional manifold with a canonical left T m -action, whose isotropy subgroups are precisely the subgroups T (p). Construction (3.1) may equally well be applied to the positive cone Rm , in which case the result is the complex vector space Cm . Since the embedding iP of (2.6) respects facial codimensions, there is a pullback diagram ZP - - Cm -Z -
i i

(3.2)

P

P - - Rm -P - of identification spaces. Here (z1 , . . . , zm ) is given by (|z1 |2 , . . . , |zm |2 ), the vertical maps are pro jections onto the quotients by the T m -actions, and iZ is a T m -equivariant embedding. It is sometimes convenient to rewrite Cm as R2m ,

8

VICTOR M BUCHSTABER, TARAS E PANOV, AND NIGEL RAY

in which case we substitute qj + irj for the j th coordinate zj , and let T act by rotation. Then Proposition 2.8 and Diagram (3.2) imply that iZ embeds ZP in R2m as the space of solutions of the m - n quadratic equations
m

(3.3)
k=1

2 2 cj,k qk + rk - b

k

= 0,

for 1

j

m - n.

In Lemma 4.1, we will confirm that ZP is a framed submanifold of R2m . The spaces ZP are of considerable independent interest in toric topology, having originated in [5]. They arise in homotopy theory as homotopy colimits [9], in symplectic topology as level surfaces for the moment maps of Hamiltonian torus actions, and in the theory of arrangements as complements of coordinate subspace arrangements. Details are given in [2], where they play a central r^ e as moment-angle complexes; relationships with combinatorial geometry and ol commutative algebra are also developed, and topological invariants described. In order to construct toric manifolds over P , we need one further set of data. This consists of a homomorphism : T m T n , satisfying Davis and Januszkiewicz's independence condition, namely (3.4) Fj1 · · · F
jk

is a face of codimension k

=

is monic on
s

Tjs .

Any such is called a dicharacteristic in [4]; the condition (3.4) ensures that the kernel K ( ) of is isomorphic to an (m - n)dimensional subtorus of T m . Wherever possible we abbreviate K ( ) to K . We write the subcircle (Ti ) < T n as T (Fi ), and denote the subgroup T (Fi )
Hi p

T

n

by S (p); it is, of course, the image of T (p) under . So (3.4) implies, for example, that S (w) = T n for any vertex w of P . When applied to the initial vertex v , (3.4) ensures that the restriction of to T1 в · · · в Tn is an isomorphism. The homomorphism of Lie algebras induced by may therefore be represented by an integral matrix of the form 1 0 . . . 0 1,n+1 . . . 1,m 0 1 . . . 0 2,n+1 . . . 2,m (3.5) L = . . .. . . . , .. .. . . . . . .. . . . 0 0 ... 1
n,n+1

...

n,m

in terms of appropriate coordinates for T . Since K acts freely on ZP there is a principal K -bundle : ZP M , whose base space is a 2ndimensional manifold. By construction, M may be expressed as the identification space (3.6) T n в P / where (s1 , p) (s2 , p) if and only if s-1 s2 S (p). Furthermore, M admits a 1 canonical T n -action , which is locally isomorphic to the standard action on Cn , and has quotient map : M P . The fixed points of lie over the vertices of

n

SPACES OF POLYTOPES AND COBORDISM

9

P . The construction identifies a neighbourhood of the fixed point -1 (v ) with Cn , on which the action of T n is precisely standard. Note that · = P as maps ZP P . The quadruple (M , , , P ) is an example of a toric manifold , as defined by Davis and Januszkiewicz. Any manifold with a similarly well-behaved torus action over P is equivariantly diffeomorphic to one of the form (3.6). In this sense, M is typical, and we follow the lead of [4] in working with (3.6) as our arbitrary toric manifold. Additional structure on M is associated to the facial submanifolds Mi , defined as the inverse images of the facets Fi under . It is clear that each Mi has codimension 2, and we may check that its isotropy subgroup is T (Fi ) < T n . Furthermore, the quotient map (3.7) ZP вK Ci - M

defines a canonical complex line-bundle i , whose restriction to Mi is isomorphic to the normal bundle i of its embedding in M . As explained in [5], the bundles i play an important part in understanding the integral cohomology ring of M . If ui denotes the first Chern class c1 (i ) in H 2 (M ), then H (M ) is generated multiplicatively by u1 , . . . , um , with two sets of relations. The first are monomial, and arise from the Stanley-Reisner ideal of P ; the second are linear, and arise from the matrix form (3.5) of the dicharacteristic. The latter may be expressed as (3.8) ui = -
i,n+1 un+1

- ... -

i,m um

for 1

i

n.

The following example is straightforward, but will be used in later sections. Example 3.9. Let the polytope P be the ncube I n , where I denotes the unit interval [0, 1] R. It has defining half-spaces (3.10) Hi = {x : -xi 0} for 1 i {x : xi 1} for n + 1 n i ,

2n

so the vertices are the binary sequences ( 1 , . . . , n ), where i = 0 or 1. The cube is strongly ordered by (3.10), with initial vertex the origin. Then iP embeds I n in R2n by iP (x) = (x1 , . . . , xn , 1 - x1 , . . . , 1 - xn ), and ZP is the product of unit 3spheres (S 3 )n (C2 )n . The dicharacteristic is specified by the n в 2n matrix (In : In ), and its kernel K is the ntorus {(t1 , . . . , tn , t
-1 1

,...,t

-1 n

)} < T

2n

.

So M is the product of 2spheres (S 2 )n , and (ei1 , . . . , ein ) T n acts on the 2 ith factor Si as rotation by i . The facial bundles are 1 , . . . , n , 1 , . . . , n , 2 where i denotes the Hopf line bundle over Si . The integral cohomology ring of M is generated by the 2dimensional elements ui for 1 i m, and the relations (3.8) give ui = -un+i . The Stanley-Reisner relations then reduce to u2 = 0 for all i. i

10

VICTOR M BUCHSTABER, TARAS E PANOV, AND NIGEL RAY

4. Stably complex structures, orientations, and framings On a smooth manifold N of dimension d, a stably complex structure is an equivalence class of real 2k plane bundle isomorphisms (N ) R2k-d , = where denotes a fixed GL(k , C)-bundle and k is suitably large. Two such isomorphisms are equivalent when they agree up to stabilisation; or, alternatively, when the corresponding lifts to BU of the classifying map of the stable tangent bundle of N are homotopic through lifts. In this section we identify the geometric data required to induce such structures on toric manifolds. According to [4], an omniorientation of a toric manifpld M consists of a choice of orientation for each i ; since the dicharacteristic determines a complex structure on each i , it encodes equivalent information. In [2], a choice of orientation for M is added to the definition, since no such choice is implied by . We adopt the latter convention henceforth, and refer to the dicharacteristic and the orientation of the omniorientation as necessary. An interior point of P admits an open neighborhood U , whose inverse image under the pro jection is canonically diffeomorphic to T n в U as a subspace of M . Since T n is oriented by the standard choice of basis, the orientations of P correspond to the orientations of M . We shall therefore take the view that an omnioriented toric manifold over P is equivalent to a dicharacteristic and an orientation of P . A strong ordering o also induces an orientation on P , by determining an affine isomorphism between a neighbourhood of the initial vertex and the positive cone Rn . This is independent of the orientation of any omniorientation on M . In order to explain the stably complex structure induced on M , it is convenient to study the embedding iZ of (3.2) in more detail. Lemma 4.1. The embedding iZ : ZP R2m is T m -equivariently framed by any choice of matrix (ci,j ) for the transformation Co of (2.7). Proof. We describe iZ by the m - n quadratic equations (3.3) over P Rm . At each point (q1 , r1 , . . . , qm , rm ) ZP , the m - n associated gradient vectors are given by (4.2) 2 (cj,1 q1 , cj,1 r1 , . . . , cj,m qm , cj,m rm ) for 1 j m - n,

and so form the rows of the (m - n) в q 1 r1 . . . R = . . . 00

2m matrix 2(ci,j )R, where ... 0 0 . . .. . .. . . ... q
m

r

m

is m в 2m. By definition of iP , the set of integers j1 ,. . . , jk with the property that qj1 = rj1 = · · · = qjk = rjk = 0 at some point z ZP corresponds to an intersection Fj1 · · · Fjk of facets forming a face of P of codimension k . Lemma 2.10 then applies to show that the matrix obtained by deleting the columns cj1 , . . . , cjk of (ci,j ) has rank m - n. It follows that 2(ci,j )R has rank m - n, and therefore that the gradient vectors (4.2) are linearly independent at z , and so frame iZ .

SPACES OF POLYTOPES AND COBORDISM

11

Furthermore, each of the gradient vectors (4.2) frames the corresponding quadratic hypersurface in R2m , and is T m -invariant. Remark 4.3. Lemma 4.1 provides an alternative to [4, Proposition 3.4], which gives insufficient detail for readers to complete the proof. Combining the details of Lemma 4.1 with [4, Theorem 6.11] yields an interesting extension of the latter. Theorem 4.4. Every complex cobordism class may be represented by the quotient of a free torus action on a real quadratic complete intersection. It is particularly illuminating to describe the framing of iZ in terms of analogous polytopes, as follows. Factoring out by the action of T m yields a framing of the embedding iP , and therefore of P as a submanifold with corners of Rm . Under the identification (2.5), the framing vectors may be represented by m - n independent 1parameter families of polytopes analogous to P . These families are made explicit by applying the differential d P to the rows of the matrix 2(ci,j )R. At the point (q1 , r1 , . . . , qm , rm ) in ZP , the matrix of d P is given by 2R, so the framing vectors are the rows of the (m - n) в m matrix 4(ci,j )RRt . When (ci,j ) takes the form (2.9), we may take the j th framing vector to be fj = (-an+j,1 y1 , . . . , -an+
j,n yn

, 0, . . . , 0, y

n+j

, 0, . . . , 0)

at y = iP (x), for 1 j m - n. Applying (2.5), we conclude that the corresponding 1parameter family of polytopes P (fj , t) (for -1 t 1) is obtained from P by: retaining the origin at x, rescaling Hk by -an+j,k t for 1 k n, fixing every facet opposite the initial vertex except Hn+j , and rescaling the latter by t. It is possible to reverse this procedure, and begin with a framing of iP . The corresponding T m -equivariant framing of iZ is then recovered by applying the contruction (3.1). Since P is contractible, all framings of iP are equivalent, and their lifts to iZ are equivariantly equivalent. In particular, the equivalence class of the framings described in Lemma 4.1 does not depend on the choice of strong ordering on P . We may now return to the tangent bundle (M ) of M . Our analysis is nothing more than a special case of a proof of Szczarba [11, (1.1)], and replaces that given in [4, (3.9)] which takes no account of the orientation on M . Prop osition 4.5. Any omnioriented toric manifold admits a canonical stably complex structure, which is invariant under the T n -action. Proof. Following Szczarba, there is a K -equivariant decomposition (ZP ) (iZ ) ZP в Cm , = obtained by restricting the tangent bundle (Cm ) to ZP . Factoring out K yields (4.6) (M ) ( /K ) ( (iZ )/K ) ZP вK Cm , = where denotes the (m - n)plane bundle of tangents along the fibres of . The right-hand side of (4.6) is isomorphic to m i as GL(m, C)-bundles. i=1

12

VICTOR M BUCHSTABER, TARAS E PANOV, AND NIGEL RAY

Szczarba [11, 6.2)] identifies /K with the adjoint bundle of , which is trivial because K is abelian; (iZ )/K is trivial by Lemma 4.1. So (4.6) reduces to an isomorphism (M ) R
2(m-n)

1 . . . m , =

although different choices of trivialisations may lead to different isomorphisms. Since M is connected and GL(2(m - n), R) has two connected components, such isomorphisms are equivalent when and only when the induced orientations agree on R2(m-n) . We choose the orientation which is compatible with those on (M ) and 1 . . . m , as given by the omniorientation. The induced structure is invariant under the action of T n , because iZ is m -equivariant. T The complex cobordism classes represented by the two choices of orientation in Proposition 4.5 differ by sign. The underlying smooth structure is also T n invariant, and is identical to that induced by Lemma 4.1.

5. Connected Sums In this section we review the construction of the connected sum of omnioriented toric manifolds M and M , bearing in mind from Proposition 4.5 that we must incorporate the orientations. We omitted this requirement in [4], and we deal with it here in terms of certain signs associated to the vertices of P . We assume throughout that the omniorientations on M and M consist of dicharacteristics and , corrresponding to matrices L and L of the form (3.5); and of orientations on the manifolds themselves, or equivalently, on P and P . We let P and P be strongly ordered by o and o , with initial vertices v and v respectively. The connected sum P #v ,v P may be described informally as follows. First construct the polytope Q by deleting the interior of the vertex figure Pv from P ; so Q has one new facet (v ) (which is an (n - 1)simplex), whose incident facets are ordered by o . Then construct the polytope Q from P by the same procedure. Finally, glue Q to Q by identifying (v ) with (v ), in such a way that the j th facet of Q combines with the j th facet of Q to give a single new facet for each 1 j n. The gluing is carried out by applying appropriate pro jective transformations to Q and Q . More precise details are given in [4, §6]. Note that P #v ,v P inherits no strong ordering, because v and v disappear during its formation. The combinatorial type of the connected sum may be altered, for example, by choosing alternative strong orderings on P and P . When the choices are clear, or their effect on the result is irrelevant, we use the abbreviation P # P . The face lattice LF (P # P ) is obtained from LF (P ) LF (P ) by identifying the j th facets of each, for 1 j n. In particular, (5.1) q (P # P ) = q (P ) + q (P ) - 2 and m(P # P ) = m(P ) + m(P ) - n.

SPACES OF POLYTOPES AND COBORDISM

13

By definition, the structed over P #v ,v ciated to the matrix 1 0 (5.2) L# = . . .

connected sum M #v ,v M is the toric manifold conP using the dicharacteristic # : T m +m -n T n asso
1,n+1 2,n+1

0 . . . 0 1,n+1 . . . 1,m 1 . . . 0 2,n+1 . . . 2,m . .. . . . .. . . . .. . . . . . 0 0 . . . 1 n,n+1 . . . n,m

. . .
n,n+1

. . . 1,m . . . 2,m . .. . . . . . . n,m

.

It is diffeomorphic to the equivariant connected sum of M and M at their initial fixed points. If M and M are omnioriented, the construction shows that the only obstruction to defining a compatible omniorientation on M # M is given by the orientations. We deal with this issue in Proposition 5.4 below. We write p : M # M M and p : M # M M for the maps collapsing the connected sum onto its constituent manifolds. We recall from [8] that an omniorientation associates a sign (w) to every vertex w of the quotient polytope P (or, equivalently, to every fixed point of M ). If w is the intersection of facets Fi1 · · · Fin , then (w) = ±1 measures the difference between the orientations induced on the tangent space at w by the dicharacteristic and the orientation of M respectively. The former is determined by the sum of line bundles i1 . . . in , and (w) is given by the Chern number (5.3) (w) = ui1 · · · uin , µ
M

,

where µM denotes the fundamental class in H2n (M ) corresponding to the orientation of M . Prop osition 5.4. The connected sum M #v ,v M admits an orientation compatible with those of M and M if and only if (v ) = - (v ). Proof. The facets of P # P give rise to complex line bundles i , j and k over M # M , corresponding to the columns of (5.2). We denote their first Chern classes by c1 (i ) = wi , c1 (j ) = wj , and c1 (k ) = wk in H 2 (M # M ), for 1 wi = - i n, n+1 j m, and n + 1 k m

respectively. The relations (3.8) become (5.5)
i,n+1

wn+1 - . . . -

i,m

wm -

i,n+1

wn+1 - . . . - n.

i,m

wm ,

which imply that wi = p ui + p u
i

for 1

i

Since the first n facets of P # P do not define a vertex, it follows that w1 · · · wn = 0 in H 2n (M # M ), and (p u1 + p u1 ) · · · (p un + p un ) = p (u1 · · · un ) + p (u1 · · · un ) = 0. For any choice of fundamental class in H2n (M # M ), we deduce that u1 · · · un , p µM
#M

+ u1 · · · un , p µM

#M

= 0.

14

VICTOR M BUCHSTABER, TARAS E PANOV, AND NIGEL RAY

But the corresponding orientation of M # M is compatible with those of M and M if and only if p µM #M = µM and p µM #M = µM ; that is, if and only if (v ) + (v ) = 0, as required. Corollary 5.6. Let M and M be omnioriented toric manifolds ordered polytopes P and P respectively, with (v ) + (v ) = 0; t complex structure induced on M #v ,v M by Proposition 4.5 tion 5.4 is equivalent to the connected sum of those induced on Moreover, the associated complex cobordism classes satisfy [M #
v ,v

over strongly hen the stably and ProposiM and M .

M ] = [M ] + [M ].

Proof. The stably complex structures on M and M combine to give an isomorphism (5.7) (M # M ) R
2(m +m -n)

1 . . . n = n+1 . . . m n+1 . . . m .

As explained in [4, Theorem 6.9], the isomorphism (5.7) belongs to one of the two equivalence classes specified by Proposition 4.5 over M # M . The choice of orientation is then provided by Proposition 5.4. The equation of cobordism classes follows immediately, because the connected sum is cobordant to the disjoint union. Proposition 5.4 implies that we cannot always form the connected sum of two omniorented toric manifolds. If the sign of every vertex of P is positive, for example, then it is impossible to construct M # M directly; we illustrate this situation in Example 6.1. Such restrictions are vital in making applications to complex cobordism theory. Corollary 5.6 confirms that the complex cobordism class [M #v ,v M ] is independent of the strong orderings o and o , and therefore of the initial vertices. Example 5.8. In example 3.9, an omniorientation is defined on S = (S 2 )n by investing the cube I n with its natural orientation as a submanifold of Rn . The stably complex structure induced by Proposition 4.5 takes the form (S ) R
2n

1 · · · n =1 n

2 This structure bounds because i i is trivial over Si for every i, and extends 3 over the disc Di . The signs of the vertices are given by

( 1, . . . ,

n

) = (-1) 1 . . . (-1) n ,

so those of adjacent vertices are opposite. We are now in a position to illustrate our principal philosophical point; that good alternative representatives can be chosen for a complex cobordism class [M ], when M itself is an omnioriented toric manifold which is not amenable to forming connected sums.

SPACES OF POLYTOPES AND COBORDISM

15

Lemma 5.9. Let M be an omnioriented toric manifold of dimension 4, over a strongly ordered polytope P ; then there exists an omnioriented M over a polytope P such that [M ] = [M ] and P has vertices of opposite sign. Proof. Suppose that v is the initial vertex of P . Let S be the omnioriented product of 2spheres given by example (3.9), with initial vertex w. If (v ) = -1, define M to be S #v,w M over the polytope P = I n #v,w P . Then [M ] = [M ], because S bounds; moreover, adjacent pairs of noninitial vertices of I n have opposites signs, which survive under the formation of P , as required. If (v ) = +1, we make the same construction using the opposite orientation on I n (and therefore on S ). Since -S also bounds, the same conclusions hold. We may now complete the proof of our amended [4, Theorem 6.11]. Theorem 5.10. In dimensions > 2, every complex cobordism class contains a toric manifold, necessarily connected, whose stably complex structure is induced by an omniorientation, and is therefore compatible with the action of the torus. Proof. Following [4], we consider cobordism classes [M1 ] and M2 ], represented by 2n-dimensional omnioriented toric manifolds over quotient polytopes P1 and P2 respectively, It suffices to construct a third such manifold M , over a quotiemt polytope P , whose cobordism class is [M1 ] + [M2 ]. To achieve this aim, we replace M2 by M2 over P2 following Lemma 5.9. We are then guaranteed to be able to construct M1 # M2 over P1 # P2 , using appropriate strong orderings on P1 and P2 . The omniorientation on M1 # M2 defines the required cobordism class, by Corollary 5.6 and Lemma 5.9. We refer to the polytope P of Theorem 5.10 as the box sum P1 P2 of P1 and P2 , because it is constructed by connecting them with an itermediate cube. The following observation of [4] is unaffected: for any complex cobordism class, the representing toric manifold may be chosen so that its quotient polytope is a connected sum of products of simplices. 6. Examples and concluding remarks We were taught the importance of adding an orientation to the original definition of omniorientation by certain 4dimensional examples of Feldman [6]. In this section we describe and develop his examples (noting that 4 is the smallest dimension to which Proposition 5.4 is relevant). They lead to our concluding remarks concerning higher dimensions. Example 6.1. The complex pro jective plane CP 2 admits a standard omniorientation, arising from its structure as a complex pro jective toric variety. The polytope P is the standard 2simplex (2), strongly ordered by the standard basis for R2 , with initial vertex at the origin. Then ZP is the unit sphere S 5 C3 . The dicharacteristic is specified by the 2 в 3 matrix 1 0 -1 , and its 0 1 -1 kernel K is the diagonal subcircle T = {(t, t, t)} < T 3 .

16

VICTOR M BUCHSTABER, TARAS E PANOV, AND NIGEL RAY

So T 3 /T is isomorphic to T 2 , and (t1 , t2 ) acts on [z1 , z2 , z3 ] CP 2 to give [t1 z1 , t2 z2 , z3 ]. Every facial bundle is isomorphic to . The integral cohomology ring of M is generated by 2dimensional elements u1 , u2 , u3 , and the relations (3.8) give u1 = u2 = u3 ; the Stanley-Reisner relations reduce to u3 = 0. Every 1 vertex of (2) has sign +1. The complex cobordism class [CP 2 ] is an additive generator of the cobordism U group 4 Z2 , which immediately raises the question of representing 2[CP 2 ]. = This is not, however, possible by omniorienting CP 2 # CP 2 , because no vertices of opposite sign are available in (2), as demanded by Proposition 5.4. Instead, we appeal to Lemma 5.9, and replace the second CP 2 by the omnioriented toric manifold (-S ) # CP 2 over P = I 2 # (2). Of course (-S ) # CP 2 is cobordant to CP 2 , and P is a pentagon. These observations lead naturally to our second example. Example 6.2. The omnioriented toric manifold CP 2 # (-S ) # CP 2 represents 2[CP 2 ], and lies over the box sum (2) (2), which is a hexagon. Figure 1 illustrates the procedure diagramatically, in terms of dicharacteristics and orientations. Every vertex of the hexagon has sign 1.
(0,1)

(0,1) ' (-1,-1) 4 4 4 4 ,0) (1 4 4

#

d (1,0) (0,1) 'd d d d d (1,0) (0 ,1) d d

#

4 (1,0) 4 4 (- 4 ' 1 4 4 (0,1)

(1,0)

,-1)

-1,-1) ' (

=
(-1,-1)

(1,0)

1) (0,

Figure 1. The omnioriented connected sum CP 2 # (-S ) # CP 2 . Our analysis is supported by a result of [8], which identifies the top Chern number of any 2ndimensional omnioriented toric manifold as (6.3) cn (M ) =
w

(w).

Given the quotient polytope P , it is convenient to refine the notation of (5.1) by writing q (P ) = q+ (M ) + q- (M ), where q± (M ) denotes the number of vertices with sign ±1 respectively. Then (6.3) shows that q( M ) - q- (M ) is a cobordism invariant of M . This is illustrated by Example 6.1, for which c2 (CP 2 ) = 3 and q- (CP 2 ) = 0. It follows by additivity that c2 (M ) = 6 for any omnioriented toric manifold representing 2[CP 2 ]; and therefore that the quotient polytope has 6 or more vertices. In particular, as observed by Feldman, M cannot be constructed over an oriented copy of (2) # (2), which is a square! An independent additive generator of 4 is represented by (CP 1 )2 , which has second Chern number 4, and may certainly be realised over the square.

SPACES OF POLYTOPES AND COBORDISM

17

Our third example shows a related 4dimensional situation in which the connected sum of the quotient polytopes does support a suitable orientation. Example 6.4. Let CP denote the underlying toric manifold of (6.1), in which the dicharacteristic is unaltered but the orientation of (2) is reversed. There2 fore every vertex has sign -1, and we may construct CP 2 # CP as an omnioriented toric manifold over (2) # (2). Figure 2 illustrates the procedure diagramatically, in terms of dicharacteristics and orientations.
(0,1) 2

(0,1) ' (-1,-1) # 4 4 4 4 4 4 (1,0) 4 4
(0,1)

4 4 4 4 (- 1 4' ,-1) 4 4 4 = (1,0) (- '
1,-1) (1,0) (-1,-1)

Figure 2. The omnioriented connected sum CP 2 # CP 2 .
U Of course [CP ] = -[CP 2 ]. So [CP 2 ] + [CP ] = 0 in 4 , and the resulting manifold bounds by Proposition 5.6. 2 2

One other observation on 2dimensional box sums is also worth making. Given k - and k -gons P and P in R2 , it follows from (5.1) that q (P P ) = q (P ) + q (P ) and m(P P ) = m(P ) + m(P ). Thus q (P P ) = m(P P ) = k + k . So P P is a (k + k )-gon, and is combinatorially equivalent to the Minkowski sum P + P whenever P and P are in general position. A situation similar to that of Example 6.2 arises in higher dimensions, when we consider the problem of representing complex cobordism classes by smooth pro jective toric varieties. For any such V , the top Chern number coincides with the Euler characteristic, and is therefore equal to the number of vertices of the quotient polytope P ; so q- (V ) = 0, by (6.3). Moreover, the Todd genus satisfies Td (V ) = 1. Remarks 6.5. Suppose that smooth pro jective toric varieties V1 and V2 are of dimension 4, and have quotient polytopes P1 and P2 respectively. Then cn (V1 ) = q (P1 ) and cn (V2 ) = q (P2 ), yet q (P1 # P2 ) = q (P1 ) + q (P2 ) - 2, from (5.1). Since cn is additive, no omnioriented toric manifold over P1 # P2 can possibly represent [V1 ] + [V2 ]. This ob jection vanishes for P1 P2 , because it enjoys an additional 2n - 2 vertices. The fact that no smooth pro jective toric varienty can represent [V1 ] + [V2 ] follows immediately from the Todd genus. Example 6.6. For any non-negative integers r and s such that r + s > 0, the cobordism class r[CP 2 ] + s[CP 1 ]2 is represented by an omnioriented toric

18

VICTOR M BUCHSTABER, TARAS E PANOV, AND NIGEL RAY

manifold M (r, s). Its quotient polytope is the iterated box sum P (r, s) =
r

(2)

s

I2 ,

which satisfies q- (P (r, s) = 0. Applying the Todd genus once more, we deduce that M (r, s) cannot be cobordant to any smooth toric variety, so long as (r, s) = (1, 0) or (0, 1). Higher dimensional examples of this phenomena are given by ... These examples suggest that we might study omnioriented toric manifolds for which q- (M ) = 0, as a natural generalisation of smooth pro jective toric varieties. Indeed, by considering q- in more detail, we may ask further and deeper questions about the representability of cobordism classes. Remarks 6.7. Suppose that an omnioriented toric manifold L has quotient polytope Q, and that its associated stably complex structure bounds. Since cn (L) = 0, it follows from (6.3) that q (Q) is even, and that half the vertices have sign +1, and half -1. We may then construct a generalised version of our connected sum, by forming the omnioriented toric manifold M1 # L # M2 over P1 # Q # P2 . Interesting possibilities for L include the bounded flag manifolds Bn of [4], in which case Q is also I n , and products such as S 2 в CP n-1 , in which case Q is I в (n - 1). We may further generalise the procedure by connecting three or more pMj over appropriate vertices of Q. Mention q- here ... References
[1] Alexander D Alexandrov. On the theory of mixed volumes of convex bodies, II (Russian). Matem Sbornik, 2(6):12051238, 1937. [2] Victor M Buchstaber and Taras E Panov. Torus Actions and Their Applications in Topology and Combinatorics. Volume 24 of University Lecture Series, Amer Math Soc, Providence, RI, 2002. [3] Victor M Buchstaber and Nigel Ray. Toric Manifolds and Complex Cobordism (Russian). Uspekhi Mat Nauk 53(2):139140, 1998. English translation in Russian Math Surveys 53(2):371373, 1998. [4] Victor M Buchstaber and Nigel Ray. Tangential structures on toric manifolds, and connected sums of polytopes. Internat Math Res Notices, 4:193219, 2001. [5] Michael W Davis and Tadeusz Januszkiewicz. Convex polytopes, Coxeter orbifolds and torus actions. Duke Math J, 62(2):417451, 1991. [6] Kostya Feldman. Private communication, 2002. [7] Peter McMullen. On simple polytopes. Inventiones Math, 113:419444, 1993. [8] Taras E Panov. Hirzebruch genera of manifolds with torus action (Russian). Izv Ross Akad Nauk Ser Mat, 65(3):123138, 2001. English translation in Izvestiya. Math, 65(3):543556, 2001. [9] Taras Panov, Nigel Ray and Rainer Vogt. Colimits, Stanley-Reisner algebras, and loop spaces. In Categorical Decomposition Techniques in Algebraic Topology, volume 215 of Progress in Mathematics. BirkhЁ r Verlag, Basel, Switzerland, 2003, pp. 261291. ause [10] Alexandr V Pukhlikov and Askold G Khovanskii. Finitely additive measures of virtual polyhedra (Russian). Algebra i Analiz, 4(2): 161185, 1992. English translation in St. Petersburg Math J, 4(2):337356, 1993. [11] R H Szczarba. On tangent bundles of fibre spaces and quotient spaces. Amer J Math, 86:685697, 1964.

SPACES OF POLYTOPES AND COBORDISM

19

Steklov Mathemtical Institute, Russian Academy of Sciences, Gubkina Street 8, 1119991 Moscow, Russia E-mail address : buchstab@mendeleevo.ru Department of Mathematics and Mechanics, Moscow State University, Leninskie Gory, 119992 Moscow, Russia E-mail address : tpanov@mech.math.msu.su School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England E-mail address : nige@ma.man.ac.uk