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Contemp orary Mathematics

Torus Actions Determined by Simple Polytop es
Victor M. Buchstaber and Taras E. Panov
Abstract. An n-dimensional polytope P n is called simple if exactly n codimension-one faces meet at each vertex. The lattice of faces of a simple polytope P n with m codimension-one faces defines an arrangement of coordinate subspaces in Cm . The group Rm-n acts on the complement of this arrangement by dilations. The corresponding quotient is a smooth manifold ZP invested with a canonical action of the compact torus T m with the orbit space P n . For each smooth pro jective toric variety M 2n defined by a simple polytope P n with the given lattice of faces there exists a subgroup T m-n T m acting freely on ZP such that ZP /T m-n = M 2n . We calculate the cohomology ring of ZP and show that it is isomorphic to the cohomology of the Stanley-Reisner ring of P n regarded as a module over the polynomial ring. In this way the cohomology of ZP acquires a bigraded algebra structure, and the additional grading allows to catch combinatorial invariants of the polytop e. At the same time this gives an example of explicit calculation of the cohomology ring for the complement of a subspace arrangement defined by simple polytope, which is of independent interest.

Intro duction A convex n-dimensional polytope P n is called simple if exactly n codimensionone faces meet at each vertex. Such polytopes are generic points in the variety of all n-dimensional convex polytopes. One can associate to each simple polytope P n a smooth (m + n)-dimensional manifold ZP with canonical action of the torus T m on it; here m is the number of codimension-one faces of P n . A number of manifolds playing an important role in different aspects of topology, algebraic and symplectic geometry are quotients ZP /T k for the action of some subgroup T k T m . The most well-known class of such manifolds are the (smooth, pro jective) toric varieties in algebraic geometry. From the viewpoint of our approach, these toric varieties (or toric manifolds) correspond to those simple polytopes P n for which there exists a torus subgroup in T m of maximal possible rank m - n that acts freely on ZP . Thus, all toric manifolds can be obtained as quotients of ZP by a torus of above type.
1991 Mathematics Subject Classification. Primary 57R19, 57S25; Secondary 14M25, 52B05. Partially supported by the Russian Foundation for Fundamental Research, grant no. 99-0100090, and INTAS, grant no. 96-0770.
c 0000 (copyright holder)

1


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VICTOR M. BUCHSTABER AND TARAS E. PANOV

The manifolds ZP were firstly introduced in [DJ] via certain equivalence relation as ZP = T m × P n / (see Definition 1.2). We propose another approach to defining ZP based on a construction from the algebraic geometry of toric varieties. This construction was initially used in [Ba] (see also [Au], [Co]) for definition of toric manifolds. Namely, the combinatorial structure of P n defines an algebraic set U (P n ) Cm (the complement of a certain arrangement of coordinate subspaces, see Definition 2.1) with action of the group (C )m on it. Toric manifolds appear when one can find a subgroup D (C )m isomorphic to (C )m-n that acts freely on U (P n ). However, it turns out that it is always possible to find a subgroup R (C )m isomorphic to Rm-n that acts on U (P n ) freely. Then, for each subgroup R of such kind the corresponding quotient U (P n )/R is homeomorphic to ZP . One of our main goals here is to study relationships between the combinatorial structure of simple polytopes and the topology of the above manifolds. One aspect of this relation is the existence of a certain bigraded complex calculating the cohomology of ZP . This bigraded complex arises from the interesting geometric structure on ZP , which we call bigraded cel l structure. This structure is defined by the torus action and the combinatorics of polytope. Thus, it seems to us that the above manifolds defined by simple polytopes could be also used as a powerful combinatorial tool. Part of results of this article were announced in [BP1]. The authors express special thanks to Nigel Ray, since the approach to studying the manifolds ZP described here was partly formed during the work on the article [BR]. 1. Manifolds defined by simple p olytop es We start with reviewing some basis combinatorial ob jects associated with simple polytopes. The good references here are [Br] and [Zi]. For any simple P n , let fi denote the number of faces of codimension (i + 1), 0 i n - 1. The integer vector (f0 , . . . , fn-1 ) is called the f -vector of P n . It is convenient to set f-1 = 1. We will also consider the another integral vector (h0 , . . . , hn ) called h-vector of P n , where hi are retrieved from the formula h0 tn + . . . + h that is,
n n n-1

t + hn = (t - 1)n + f0 (t - 1)

n-1

+ ... + f

n-1

,

(1.1)
i=0

hi tn

-i

=
i=0

f

i-1

(t - 1)

n-i

.

This implies that
k

(1.2)

hk =
i=0

(-1)k

-i

n-i f k-i

i-1

.

n n Now let F = (F1 -1 , . . . , Fm-1 ) be the set of all codimension-one faces of P n , so m = f0 . We fix a commutative ring k, which we refer to the as ground ring.

Definition 1.1. The face ring (or the Stanley-Reisner ring) k(P n ) is defined to be the ring k[v1 , . . . , vm ]/I , where I = (vi1 . . . v
i
s

:

i 1 < i 2 < . . . < is ,

Fi1 Fi2 ž ž ž Fis = ) .


TORUS ACTIONS DETERMINED BY SIMPLE POLYTOPES

3

Thus, the face ring is a quotient ring of polynomial ring by an ideal generated by some square free monomials of degree 2. We make k(K ) a graded ring by setting deg vi = 2, i = 1, . . . , m. For any simple polytope P n one can define (n - 1)-dimensional simplicial complex KP dual to the boundary P n . Originally, face ring was defined by Stanley [St2] for simplicial complexes. In our case Stanley's face ring k(KP ) coincides with k(P n ). Below for any simple P n with m codimension-one faces we define, following [DJ], two topological spaces ZP and BT P . Set the standard basis {e1 , . . . , em } in Zm , and define canonical coordinate subgroups Tik ,...ik T m as tori corresponding to the sublattices spanned in Zm by 1 ei1 , . . . , eik . Definition 1.2. The space ZP associated with simple polytope P n is ZP = (T m × P n )/ , where the equivalence relation is defined as follows: (g1 , p) - (g2 , q ) p = q and g1 g2 1 Tik ,...,ik . Here {i1 , . . . , ik } is the set of indices of al l 1 codimension-one faces containing the point p P n , that is, p Fi1 ž ž ž Fik . Note that dim ZP = m + n. The torus T m acts on ZP with orbit space P n . This action is free over the interior of P n and has fixed points corresponding to vertices of P n . It was mentioned above that there are other well-known in algebraic geometry examples of manifolds with torus action and orbit space a simple polytope. These are the toric varieties [Da], [Fu] (actually, we consider only smooth pro jective toric varieties). The space ZP is related to this as follows: for any smooth toric variety M 2n over P n the orbit map ZP P n decomposes as ZP M 2n P n , where ZP M 2n is a principal T m-n -bundle, and M 2n P n is the orbit map for M 2n . We will review this connection with more details later. Example 1.3. Let P n = n (an n-dimensional simplex). Then m = n + 1, and it is easy to check that ZP = (T n+1 × n )/ S 2n+1 . = Using the action of T struction) (1.3)
m m

on ZP , define the homotopy quotient (the Borel conBT P = E T
m

×T

m

ZP ,

where E T is the contractible space of universal T m -bundle over B T m = (CP )m . It is clear that the homotopy type of BT P is defined by the simple polytope P n . Let I q be the standard q -dimensional cube in Rq : I q = {(y1 , . . . , yq ) Rq : 0 yi 1, i = 1, . . . , q }. A cubical complex is a topological space represented as the union of homeomorphic images of standard cubes in such a way that the intersection of any two cubes is a face of each. Lemma 1.4. Any simple polytope P n has a natural structure of cubical complex n C , which has s = fn-1 n-dimensional cubes Iv indexed by the vertices v P n and 1 + f0 + f1 + . . . + fn-1 vertices. Moreover there is a natural embedding iP of C into the boundary complex of standard m-dimensional cube I m . Proof. Let us take a point in the interior of each face of P n (we also take all vertices and a point in the interior of the polytope). The resulting set S of 1 + f0 + f1 + . . . + fn-1 points is said to be the vertex set of the cubical complex


4

VICTOR M. BUCHSTABER AND TARAS E. PANOV

P

n
?

C ?e ? e

I
e

m

C G

D

? A

?

e ? e ? B ?r e Å rrÅÅ ? G ? ? F

?

B D e e e iP E

e

e e E A

E F

Figure 1. The embedding ip : P n I

m

for n = 2, m = 3.

C . Now let us construct the embedding C I m . We map the point of S that lies inside the face F n-k = Fin-1 ž ž ž Fin-1 , k 1 to the vertex of the cube I m whose 1 k yi1 , . . . , yik coordinates are zero, while other coordinates are 1. The point of S in the interior of P n is then mapped to the vertex of I m with coordinates (1, . . . , 1). Let us consider the simplicial subdivision K of the polytope P n that is constructed as the cone over the barycentric subdivision of simplicial complex K n-1 dual to the boundary of P n . The vertex set of the simplicial complex K is our set S , and for any vertex v of P n one can find a subcomplex Kv K (the cone over the barycentric subdivision of the (n - 1)-simplex in K n-1 corresponding to v ) that n simplicially subdivides the cube Iv . Now we can extend the map S I m linearly on each simplex of the triangulation K to the embedding iP : P n I m . Figure 1 describes the embedding iP : P n I m in the case n = 2, m = 3. The embedding iP : P n I m has the following property: Proposition 1.5. If v = Fin-1 ž ž ž Fin-1 is a vertex of P n , then the cube 1 n I P n is mapped onto the n-face of the cube I m determined by m - n equations yj = 1, j {i1 , . . . , in }. /
n v

Now, let us consider the standard unit poly-disk (D2 )m = {(z1 , . . . , zm ) Cm : |zi | 1} Cm . The standard action of T m on Cm by diagonal matrices defines the action of T on (D2 )m with orbit space I m .
m

Theorem 1.6. The space ZP has a canonical structure of smooth (m + n)dimensional manifold such that the T m -action is smooth. The embedding iP : P n I m constructed in Lemma 1.4 is covered by a T m -equivariant embedding ie : ZP


TORUS ACTIONS DETERMINED BY SIMPLE POLYTOPES

5

(D2 )m Cm . This can be described by the commutative diagram ZP - - (D2 )m -e - P
n i

-- -P -

i

I m,

Proof. Let : ZP P n be the orbit map. It easily follows from the definition n n of ZP that for each cube Iv P n (see Lemma 1.4) we have -1 (Iv ) (D2 )n × = m-n 2n n T . Here (D ) is the unit poly-disk in C with diagonal action of T n . Hence, n ZP is represented as the union of blocks Bv = -1 (Iv ), each of which is isomorphic 2n m- n to (D ) × T . These blocks Bv are glued together along their boundaries to get the smooth T m -manifold ZP . Now, let us prove the second part of the theorem concerning the equivariant embedding. First, we fix a numeration of codimension-one faces of P n : n n F1 -1 , . . . , Fm-1 . Take the block Bv (D2 )n × T m-n = D2 × . . . × D2 × S 1 × . . . × S 1 = corresponding to a vertex v P n . Each factor D2 or T 1 in Bv corresponds to a codimension-one face of P n and therefore acquires a number (index) i, 1 i m. Note that n factors D2 acquire the indices corresponding to those codimension-one faces containing v , while other indices are assigned to m - n factors T 1 . Now we numerate the factors D2 (D2 )m of the poly-disk in any way and embed each block Bv ZP into (D2 )m according to the indexes of its factors. It can be easily seen that the embedding of a face I n given by m - n equations of type yj = 1 (as in Proposition 1.5) into the cube I m is covered by the above constructed embedding of Bv (D2 )n × T m-n into (D2 )m . Then it follows from Proposition 1.5 that the = set of embeddings (D2 )n × T m-n Bv (D2 )m defines an equivariant embedding = ie : ZP (D2 )m . By the construction, this embedding covers the embedding iP : P n I m from Lemma 1.4. Example 1.7. If P n = 1 is an 1-dimensional simplex (a segment), then Bv = D × S 1 for each of the two vertices, and we obtain the well-known decomposition Z1 S 3 = D2 × S 1 S 1 × D2 . If P n = n is an n-dimensional simplex, we obtain = the similar decomposition of a (2n + 1)-sphere into n + 1 "blocks" (D2 )n × S 1 .
2

2. Connections with toric varieties and subspace arrangements The above constructed embedding ie : ZP (D2 )m Cm allows us to connect the manifold ZP with one construction from the theory of toric varieties. Below we describe this construction, following [Ba]. Definition 2.1. Let I = {i1 , . . . , ip } be an index set, and let AI Cm denote the coordinate subspace zi1 = ž ž ž = zip = 0. Define the arrangement A(P n ) of subspaces of Cm as A(P n ) = AI ,
I

where the union is taken over all I = {i1 , . . . , ip } such that Fi1 ž ž ž Fip = in P n . Put U (P n ) = Cm \ A(P n ).


6

VICTOR M. BUCHSTABER AND TARAS E. PANOV

Note that the closed set A(P n ) has codimension at least 2 and is invariant with respect to the diagonal action of (C )m on Cm . (Here C denote the multiplicative group of non-zero complex numbers). Hence, (C )m , as well as the torus T m (C )m , acts on U (P n ) Cm . It follows from Proposition 1.5 that the image of ZP under the embedding ie : ZP Cm (see Theorem 1.6) does not intersect A(P n ), that is, ie (ZP ) U (P n ). We put Rm = {(1 , . . . , m ) Rn : i > 0}. > This is a group with respect to multiplication, which acts by dilations on Rm and Cm (an element (1 , . . . , m ) Rm takes (y1 , . . . , ym ) Rm to (1 y1 , . . . , m ym )). > There is the isomorphism exp : Rm Rm between additive and multiplicative > groups, which takes (t1 , . . . , tm ) Rm to (et1 , . . . , etm ) Rm . > Remember that the polytope P n is a set of points x Rn satisfying m linear inequalities: (2.1) P n = {x Rn : li , x -ai , i = 1, . . . , m}, where li (Rn ) are normal (co)vectors of facets. The set of (÷1 , . . . , ÷m ) Rm such that ÷1 l1 + . . . + ÷m lm = 0 is an (m - n)-dimensional subspace in Rm . We choose a basis {wi = (w1i , . . . , wmi ) }, 1 i m - n, in this subspace and form the m × (m - n)-matrix w11 . . . w1,m-n ... (2.2) W = ... ... wm1 . . . wm,m-n of maximal rank m - n. This matrix satisfies the following property. Proposition 2.2. Suppose that n facets Fin-1 , . . . , Fin-1 of P n meet at the 1 n same vertex v : Fin-1 ž ž ž Fin-1 = v . Then the minor (m - n) × (m - n)1 n matrix Wi1 ...in obtained from W by deleting n rows i1 , . . . , in is non-degenerate: det Wi1 ...in = 0. Proof. Suppose det Wi1 ,...,in = 0, then one can find a zero non-trivial linear combination of vectors li1 , . . . , lin . But this is impossible: since P n is simple, the set of normal vectors of facets meeting at the same vertex constitute a basis of Rn . The matrix W defines the subgroup R
W

= {(e

w11 1 +žžž+w1

,m-n m-n

,...,e

wm1 1 +žžž+wm,m-n m

-n

) Rm } Rm , > >

where (1 , . . . , m-n ) runs over Rm-n . This subgroup is isomorphic to Rm-n . Since > U (P n ) Cm (see Definition 2.1) is invariant with respect to the action of Rm > (C )m on Cm , the subgroup RW Rm also acts on U (P n ). > Theorem 2.3. The subgroup RW Rm acts freely on U (P n ) Cm . The > composite map ZP U (P n ) U (P n )/RW of the embedding ie and the orbit map is a homeomorphism. Proof. A point from Cm may have the non-trivial isotropy subgroup with respect to the action of Rm on Cm only if at least one of its coordinates vanishes. > As it follows from Definition 2.1, if a point x U (P n ) has some zero coordinates, then all of them correspond to facets of P n having at least one common vertex v P n . Let v = Fin-1 ž ž ž Fin-1 . Then the isotropy subgroup of the point x 1 n


TORUS ACTIONS DETERMINED BY SIMPLE POLYTOPES

7

with respect to the action of RW is non-trivial only if some linear combination of vectors w1 , . . . , wm-n lies in the coordinate subspace spanned by ei1 , . . . , ein . But this means that det Wi1 ...in = 0, which contradicts Proposition 2.2. Thus, RW acts freely on U (P n ). Now, let us prove the second part of the theorem. Here we use both embeddings ie : ZP (D2 )m Cm from Theorem 1.6 and iP : P n I m Rm from Lemma 1.4. It is sufficient to prove that each orbit of the action of RW on U (P n ) Cm intersects the image ie (ZP ) at a single point. Since the embedding ie is equivariant, instead of this we may prove that each orbit of the action of RW on the real part UR (P n ) = U (P n ) Rm intersects the image + iP (P n ) in a single point. Let y iP (P n ) Rm . Then y = (y1 , . . . , ym ) lies n in some n-face Iv of the unit cube I m Rm as described by Proposition 1.5. We need to show that the (m - n)-dimensional subspace spanned by the vectors (w11 y1 , . . . , wm1 ym ) , . . . , (w1,m-n y1 , . . . , wm,m-n ym ) is in general position with n the n-face Iv . But this follows directly from Propositions 1.5 and 2.2. The above theorem gives a new proof of the fact that ZP is a smooth manifold, which allows a T m -equivariant embedding into Cm R2m with trivial normal = bundle. Example 2.4. Let P n = n (an n-simplex). Then m = n + 1, U (P n ) = m C \ {0}, R> -n is R> , and R> takes z Cn+1 to z . Thus, we have 2n+1 ZP = S (this could be also deduced from Definition 1.2; see also Example 1.7).
n+1

Now, suppose that all vertices of P n belong to the integer lattice Zn Rn . Such an integral simple polytope P n defines a pro jective toric variety MP (see [Fu]). Normal (co)vectors li of facets of P n (see (2.1)) can be taken integral and primitive. The toric variety MP defined by P n is smooth if for each vertex v = Fi1 . . .Fin the vectors li1 , . . . , lin constitute an integral basis of Zn . As before, we may construct the matrix W (see (2.2)) and then define the subgroup C
W

= {(e

w11 1 +žžž+w1

,m-n m-n

,...,e

wm1 1 +žžž+wm,m-n m

-n

)} (C )m ,

where (1 , . . . , m-n ) runs over Cm-n . This subgroup is isomorphic to (C )m-n . It can be shown (see [Ba]) that CW acts freely on U (P n ) and the toric manifold MP is identified with the orbit space (or the geometric quotient) U (P n )/CW . Thus, we have the commutative diagram U (P n ) - - - - - - - ZP m-n M
2n R
W

R =

m-n >

CW (C ) =

T

m-n

M

2n

.

Since ZP can be viewed as the orbit space of U (P n ) with respect to the action of RW Rm-n , the manifold ZP and the complement U (P n ) of an arrangement of => planes are of same homotopy type. In the next section we calculate the cohomology ring of ZP (or U (P n )). 3. Cohomology ring of Z
P

The following lemma follows readily from the construction of ZP .


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VICTOR M. BUCHSTABER AND TARAS E. PANOV

n n Lemma 3.1. If P n is the product of two simple polytopes: P n = P1 1 × P2 2 , n1 n then ZP = ZP1 × ZP2 . If P1 P is a face, then ZP1 is a submanifold of ZP .

The space B T m = (CP )m has a canonical cellular decomposition (that is, each CP has one cell in each even dimension). For each index set I = {i1 , . . . , ik } k we introduce the cellular subcomplex B TI = B Tik ,...,ik B T m homeomorphic 1 k to B T . Definition 3.2. Define the cellular subcomplex BT P B T m to be the union k of B TI over all I = {i1 , . . . , ik } such that Fi1 ž ž ž Fip = in P n . Theorem 3.3. The cel lular embedding i : BT P B T m (see Definition 3.2) and the fibration p : BT P B T m (see (1.3)) are homotopical ly equivalent. In particular, BT P and BT P are of same homotopy type. Proof. The proof can be found in [BP1]. Corollary 3.4. The cohomology ring of BT P is isomorphic to the face ring k(P n ) (see Definition 1.1). The projection p : BT P B T m induces the quotient epimorphism p : k[v1 , . . . , vm ] k(P n ) = k[v1 , . . . , vm ]/I in the cohomology. A simple polytope P n with m codimension-one faces is called q -neighbourly [Br] n if the (q - 1)-skeleton of the simplicial complex KP -1 dual to the boundary P n coincides with the (q - 1)-skeleton of an (m - 1)-simplex. Equivalently, P n is q neighbourly if any q codimension-one faces of P n have non-empty intersection. Note that any simple polytope is 1-neighbourly. The next theorem about the homotopy groups of ZP and BT P follows easily from cellular structure of BT P and exact homotopy sequence of the bundle p : BT P B T m with fibre ZP . Theorem 3.5. For any simple polytope P have: (1) (2) (3) (4)
n

with m codimension-one faces we

1 (ZP ) = 1 (BT P ) = 0. 2 (ZP ) = 0, 2 (BT P ) = Zm . q (ZP ) = q (BT P ) for q 3. If P n is q -neighbourly, then i (ZP ) = 0 for i < 2q + 1, and 2q+1 (ZP ) is a free Abelian group whose generators correspond to monomials vi1 ž ž ž viq+1 I (see Definition 1.1). ZP - - B T P -- p

From (1.3) we obtain the commutative square

- - BT m. -- The Eilenberg-Moore spectral sequence [Sm] of this square has the E2 -term E2 Tor =
k[v1 ,...,vm ]

k(P n ), k ,

where k(P n ) is regarded as a k[v1 , . . . , vm ]-module by means of quotient pro jection k[v1 , . . . , vm ] k[v1 , . . . , vm ]/I = k(P n ). This spectral sequence converges to the cohomology of ZP . It turns out that the spectral sequence collapses at the E2 term, and moreover, the following statement holds:


TORUS ACTIONS DETERMINED BY SIMPLE POLYTOPES

9

Theorem 3.6. Provided that k is a field, we have an isomorphism of algebras: H (ZP ) = Tork[v1 ,...,vm ] k(P n ), k . The additive structure of the cohomology is thus given by the isomorphisms H r (ZP ) =
2j -i=r 2j Tor-[i,1 ,...,v kv
m

]

k(P n ), k ,

i, j 0.

Proof. The proof of the theorem uses some results of [Sm] on the Eilenberg- Moore spectral sequences and Theorem 3.3. This proof can be found in [BP1]. Suppose that there is at least one smooth toric variety M 2n whose orbit space with respect to the T n -action has combinatorial type of the simple polytope P n . Then one can find a subgroup T m-n H T m that acts freely on the manifold = ZP such that M 2n = ZP /H . In general case such a subgroup may fail to exist; however, sometimes one can find a subgroup of dimension less than m - n that acts freely on ZP . So, let H T r be such a subgroup. Then the inclusion s : H T m = is determined by an integral (m × r)-matrix (sij ) such that the Z-module spanned by its columns sj = (s1j , . . . , smj ) , j = 1, . . . , r, is a direct summand in Zm . Choose any basis ti = (ti1 , . . . , tim ), i = 1, . . . , m - r, in the kernel of the dual map s : (Zm ) (Zr ) . Then we have the following result describing the cohomology ring of the manifold Y = ZP /H . Theorem 3.7. The fol lowing isomorphism of algebras holds: H (Y ) = Tork[t1 ,...,tm-r ] k(P n ), k , where the k[t1 , . . . , t by the map
m-r

]-module structure on k(P n ) = k[v1 , . . . , vm ]/I is defined
m- r

k[t1 , . . . , t

] k[v1 , . . . , vm ] ti ti1 v1 + . . . + t

im vm

.

Proof. This theorem, as well as the previous one, can be proved by considering a certain Eilenberg-Moore spectral sequence. See [BP1, Theorem 4.13]. In the case of toric varieties (that is, r = m - n in Theorem 3.7) we obtain H (M 2n ) = Tork[t1 ,...,tn ] k(P n ), k . It can be shown that in this case k(P n ) is a free k[t1 , . . . , tn ]-module (which implies that t1 , . . . , tn is a regular sequence, and k(P n ) is a so-called Cohen-Macaulay ring). Thus, we have H (M 2n ) k(P n )/J = k[v1 , . . . , vm ]/I +J, = where J is the ideal generated by ti1 v1 + . . . + tim vm , i = 1, . . . , n. This result (the description of the cohomology ring of a smooth toric variety) is well known in algebraic geometry as the Danilov-Jurkiewicz theorem [Da]. In order to describe the cohomology ring of ZP more explicitly, we apply some constructions from homological algebra. Let = k[y1 , . . . , yn ], deg yi = 2, be a graded polynomial algebra, and let A be any graded -module. Let [u1 , . . . , un ] denote the exterior algebra on generators u1 , . . . , un over k, and consider the complex E = [u1 , . . . , un ].


10

VICTOR M. BUCHSTABER AND TARAS E. PANOV

This is a bigraded differential algebra; its gradings and differential are defined by bideg(yi 1) = (0, 2), bideg(1 ui ) = (-1, 2), d(yi 1) = 0; d(1 ui ) = yi 1

and requiring that d be a derivation of algebras. The differential adds (1, 0) to bidegree, hence, the components E -i, form a cochain complex. It is well known that this complex is a -free resolution of k (regarded as a -module) called the Koszul resolution (see [Ma]). Thus, for any -module A we have Tor (A, k) = H A [u1 , . . . , un ], d = H A [u1 , . . . , un ], d , where d is defined as d(1 ui ) = yi 1. Applying this construction to the case = k[v1 , . . . , vm ], A = k(P n ) and using Theorem 3.6, we get the following statement. Theorem 3.8. The fol lowing isomorphism of graded algebras holds: H (ZP ) H k(P n ) [u1 , . . . , um ], d , = bideg vi = (0, 2), bideg ui = (-1, 2), d(vi 1) = 0, d(1 ui ) = vi 1,

where k(P n ) = k[v1 , . . . , vm ]/I is the face ring. Corollary 3.9. The Leray-Serre spectral sequence of the T m -bund le ZP × E T col lapses at the E3 term. Z
P m m

B T P = ZP ×

T

m

ET

Theorems 3.6 and 3.8 show that instead of usual grading, the cohomology of has bigraded algebra structure with bigraded components H
-i,2j 2j (ZP ) Tor-[i,1 ,...,v = kv
m

]

k(P n ), k ,

i, j 0,

satisfying H r (ZP ) = 2j -i=r H -i,2j (ZP ). Since ZP is a manifold, there is the Poincar? duality in H (ZP ). This Poincar? e e duality has the following combinatorial interpretation. Lemma 3.10. (1) The Poincar? duality in H e theorems 3.6 and 3.8. More class, then its Poincar? dual e (2) Let v = Fin-1 ž ž ž Fin-1 1 n . . . < jm-n , {i1 , . . . , in , j1 , . . element vi1 ž ž ž vin uj1 ž ž ZP ) regards the bigraded structure defined by precisely, if H -i,2j (ZP ) is a cohomology D belongs to H -(m-n)+i,2(m-j ) . be a vertex of the polytope P n , and let j1 < . , jm-n } = {1, . . . , m}. Then the value of the ž ujm
-n

H

m+n

(ZP )

on the fundamental class of ZP equals +1. (3) Let v1 = Fin-1 ž ž ž Fin-1 and v2 = Fin-1 ž ž ž Fin-1 Fjn-1 be two 1 n 1 n-1 vertices of P n connected by an edge, and j1 , . . . , jm-n as above. Then vi1 ž ž ž vin uj1 ž ž ž uj in H
m+n
m-n

= vi1 ž ž ž v

in

-1

vj1 uin uj2 ž ž ž uj

m-n

(ZP ).


TORUS ACTIONS DETERMINED BY SIMPLE POLYTOPES

11

Proof. The second assertion follows from the fact that the cohomology -(m-n),2m class under consideration is a generator of the module Tork[v1 ,...,vm ] k(P n ), k = m H m+n (ZP +n ) (see Theorem 3.6). To prove the third assertion we just mention that d(vi1 ž ž ž vi
n
n-1

uin uj1 uj2 ž ž ž ujm-n ) = vi1 ž ž ž vin uj1 ž ž ž u

jm

-n

- vi1 ž ž ž v

in

-1

vj1 uin uj2 ž ž ž uj

m-n

in k(P ) [u1 , . . . , um ] (see Theorem 3.8). Changing the numeration of codimension-one faces of P n and the orientation of ZP if necessary, we may assume that the fundamental cohomology class of ZP is represented by the cocycle v1 ž ž ž vn um+1 ž ž ž um k(P n ) [u1 , . . . , um ]. 4. New relations b etween combinatorics and top ology The results on the topology of manifolds defined by simple polytopes obtained in the previous sections give rise to new remarkable connections with combinatorics of polytopes. Here we discuss only few examples. Set T i = Tor-[i 1 ,...,vm ] k(P n ), k and T i,j = Tor-[i,j,...,vm ] k(P n ), k . Then kv k v1 Lemma 3.10 shows that the Poincar? duality for ZP can be shortly written as the e m following identity for the Poincar? series F (T i , t) = r=0 dimk (T i,2r )t2r of T i : e (4.1) F (T i , t) = t
2m

F (T

m-n-i 1 ,t

).

The above identity is well known in commutative algebra for so-called Gorenstein simplicial complexes (see [St2, p. 76]). A simplicial complex K with m vertices is Gorenstein over k if the face ring k(K ) is Cohen-Macaulay and dim Tork
-(m-n) [v1 ,...,vm ]

k(K ), k = 1,

where n is the maximal number of algebraically independent elements of k(K ), that is, the maximal number of vertices of simplices of K . It is known that the face ring of simplicial subdivision of a sphere S n-1 is Gorenstein (see [St2, p. 76]). In particular, our face ring k(P n ) of simple polytope P n is Gorenstein, and the maximal number of algebraically independent elements of k(P n ) equals the dimension of P n . A simple combinatorial argument (see [St2, part II, 1]) shows that for any simple polytope P n the Poincar? series F k(P n ), t can be written as follows e
n-1

F k(P n ), t = 1 +
i=0

fi t2(i+1) , (1 - t2 )i+1

where (f0 , . . . , fn-1 ) is the f -vector of P . This series can be also expressed in terms of the h-vector (h0 , . . . , hn ) (see (1.1)) as (4.2) F k(P n ), t = h0 + h1 t2 + . . . + hn t (1 - t2 )n
2n

.

On the other hand, one can also deduce the formula for F k(P n ), t from the Hilbert syzygy theorem by applying it to the minimal resolution of k(P n ) regarded as a k[v1 , . . . , vm ]-module. This formula is as follows (4.3) F k(P n ), t =
m i=0

(-1)i F (T i , t) . (1 - t2 )m


12

VICTOR M. BUCHSTABER AND TARAS E. PANOV

Combining (4.1), (4.2), and (4.3) we get (4.4) hi = h
n-i

,

i = 0, 1, . . . , n.

These are the well-known Dehn-Sommervil le equations (see, for example, [Br]) for simple (or simplicial) polytopes. Dehn-Sommerville equations also hold for any Gorenstein simplicial complex K (that is, such that the face ring k(K ) is Gorenstein, see [St2, p. 77]). Thus, we see that the algebraic duality (4.1) for Gorenstein simplicial complexes and the combinatorial Dehn-Sommerville equations (4.4) follow from the Poincar? duality for the manifold ZP . e Now, we define the bigraded Betti numbers b-i,2j as b-
i,2j 2j = dimk Tor-[i,1 ,...,v kv 2j -i=r
m

]

k(P n ), k .

Then by Theorem 3.6, br (ZP ) = b
0,0

b

-i,2j

. It is easy to check that

= 1,
m

b

- q ,2s

= 0 if 0 < s q .

Now, one can define Euler characteristics s as s =
q =0

(-1)q b-

q ,2s

,

s = 0, . . . , m

and then define the series (t) as
m

(t) =
s=0

s t2s .

It can be shown that for this series the following identity holds (4.5) (t) = (1 - t2 )m
-n

(h0 + h1 t . . . + hn t2n ).

This formula allows to express the h-vector (h0 , . . . , hn ) of a simple polytope P n in terms of the bigraded Betti numbers b-q,2r (ZP ) of the corresponding manifold ZP . At the end let us mention two more connections with well-known combinatorial results. Firstly, consider the first non-trivial MacMullen inequality for simple P n (see [Br]): h1 h2 , for n 3. Using identity (4.5), one can express the above inequality in terms of the bigraded Betti numbers b-q,2r as follows: (4.6) b3 (ZP ) = b
-1,4

(ZP )

m-n 2

,

for n 3.

Secondly, let us consider the well-known Upper Bound for the number of faces of simple polytope. In terms of the h-vector it is as follows: hi (see [Br]). Using the identity 1 1 - t2
m-n m-n+i-1 i

=
i=0

m - n + i - 1 2i t, i

and formula (4.5), we deduce that the Upper Bound is equivalent to the following inequality: (4.7) (t) 1, |t| 1.


TORUS ACTIONS DETERMINED BY SIMPLE POLYTOPES

13

It would be interesting to obtain a purely topological proof of inequalities (4.6) and (4.7). 5. Concluding remarks As we mentioned in the introduction, the confluence of ideas from topology and combinatorics that gives rise to our notion of manifolds defined by simple polytopes ascends to geometry of toric varieties. Toric geometry enriched combinatorics of polytopes by very powerful topological and algebraic-geometrical methods, which led to solution of many well-known problems. Here we mention only two aspects. The first one is counting lattice points: starting from [Da] the Riemann-Roch theorem for toric varieties and related results were used for calculating the number of lattice points inside integral polytopes (see also [Fu]). The second aspect is the famous Stanley theorem [St1] that proves the necessity of MacMullen's conjecture for the number of faces of a simple (or simplicial) polytope. The proof uses the pro jective toric variety constructed from a simple polytope with vertices in integral lattice. This toric variety is not determined by the combinatorial type of the polytope: it depends also on integral coordinates of vertices. Many combinatorial types can be realized as integral simple polytopes in such a way that the resulting toric variety is smooth; in this case the Dehn-Sommerville equations follow from the Poncar? duality for ordinary cohomology, while the MacMullen inequalities fole low from the Hard Lefschetz theorem. However, there are combinatorial types of simple polytopes that do not admit any smooth toric variety. The simplest examples are duals to the so-called cyclic polytopes of dimension 4 with sufficiently many vertices (see [DJ, Corollary 1.23]). For such polytopes Stanley's proof uses the Poincar? duality and the Hard Lefschetz theorem for intersection cohomology e of the corresponding (singular) toric variety. We mention that the Hard Lefschetz theorem for intersection cohomology is a very deep algebraic-geometrical result. Nevertheless, methods of toric geometry fail to give a proof of very natural generalization of MacMullen's conjecture to the case of simplicial spheres. The discussion of MacMullen's inequalities for simplicial spheres, Gorenstein complexes and related topics can be found in [St2]. On the other hand, our approach provides an interpretation of Dehn-Sommerville equations in terms of Poincar? duality in ore dinary cohomology for any combinatorial simple polytope and gives a topological interpretation of MacMullen's inequalities. Moreover, our methods extend naturally to simplicial spheres. In can be easily seen that the construction of manifold ZP and other constructions from our paper are equally applicable for non-polytopal simplicial spheres. Actually, an analog of ZP can be constructed for any simplicial complex. In general case this fails to be a manifold, however it still decomposes into blocks of type (D2 )q × T m-q as described in the proof of Theorem 1.6. We call this space the moment-angle complex defined by simplicial complex. As in the case of a simple polytope, the moment-angle complex is homotopically equivalent to a certain coordinate subspace arrangement determined by the simplicial complex (see Section 2). In our paper [BP3] we study topology of moment-angle complexes and calculate the cohomology rings of general coordinate subspace arrangements. References
[Au] M. Audin, The Topology of Torus Actions on Symplectic Manifolds, Progress in Mathematics 93, BirkhÅ auser, Boston Basel Berlin, 1991.


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VICTOR M. BUCHSTABER AND TARAS E. PANOV

D. A. Cox, Recent developments in toric geometry, in: Algebraic geometry (Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9-29, 1995), J. Kollar, (ed.) et al. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62 (pt.2), 389-436 (1997). [Ba] V. V. Batyrev, Quantum Cohomology Rings of Toric Manifolds, Journ? de G? ? ees eometrie Alg? ebrique d'Orsay (Juillet 1992), Ast? erisque 218, So ci? Math? ete ematique de France, Paris, 1993, pp. 9-34. [Br] A. Brønsted, An introduction to convex polytopes, Springer-Verlag, New-York, 1983. [BP1] V. M. Bukhshtaber and T. E. Panov, Algebraic topology of manifolds defined by simple polytopes (Russian), Uspekhi Mat. Nauk 53 (1998), no. 3, 195-196; English transl. in: Russian Math. Surveys 53 (1998), no. 3, 623-625. [BP2] V. M. Buchstaber and T. E. Panov, Torus actions and combinatorics of polytopes (Russian), Trudy Matematicheskogo Instituta im. Steklova 225 (1999), 96-131; English transl. in: Proceedings of the Steklov Institute of Mathematics 225 (1999), 87-120; available at http://xxx.lanl.gov/abs/math.AT/9909166. [BP3] V. M. Buchstaber and T. E. Panov, Torus actions, equivariant moment-angle complexes and coordinate subspace arrangements (Russian), Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 266 (2000); available at http://xxx.lanl.gov/abs/math.AT/9912199. [BR] V. M. Bukhshtaber and N. Ray, Toric manifolds and complex cobordisms (Russian), Uspekhi Mat. Nauk 53 (1998), no. 2, 139-140; English transl. in: Russian Math. Surveys 53 (1998), no. 2, 371-373. [Da] V. Danilov, The geometry of toric varieties, (Russian), Uspekhi Mat. Nauk 33 (1978), no. 2, 85-134; English transl. in: Russian Math. Surveys 33 (1978), 97-154. [DJ] M. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. Journal 62, (1991), no. 2, 417-451. [Fu] W. Fulton, Introduction to Toric Varieties, Princeton Univ. Press, Princeton, NJ, 1993. [Ma] S. Maclane, Homology, Springer-Verlag, Berlin, 1963. [Sm] L. Smith, Homological Algebra and the Eilenberg-Moore Spectral Sequence, Transactions of American Math. Soc. 129 (1967), 58-93. [St1] R. Stanley, The number of faces of a convex simplicial polytope, Adv. in Math. 35 (1980), 236-238. [St2] R. Stanley, Combinatorics and Commutative Algebra, Progress in Mathematics 41, Birkhauser, Boston, 1983. [Zi] G. Ziegler, Lectures on Polytopes, Graduate Texts in Math. 152, Springer-Verlag, Berlin Heidelberg New-York, 1995. Department of Mathematics and Mechanics, Moscow State University, 119899 Moscow, RUSSIA E-mail address : buchstab@mech.math.msu.su E-mail address : tpanov@mech.math.msu.su

[Co]