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Proceedings of the Steklov Institute of Mathematics, Vol. 247, 2004, pp. 1­17. Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, Vol. 247, 2004, pp. 41­58. Original Russian Text Copyright c 2004 by V.M. Buchstaber, T.E. Panov. English Translation Copyright c 2004 by MAIK "Nauka/Interperiodica" (Russia).

Combinatorics of Simplicial Cell Complexes and Torus Actions
V. M. Buchstaber1 and T. E. Panov2
Received April 2004

Abstract--Simplicial cell complexes are special cellular decompositions also known as virtual or ideal triangulations; in combinatorics, appropriate analogues are given by simplicial partially ordered sets. In this paper, combinatorial and topological properties of simplicial cell complexes are studied. Namely, the properties of f -vectors and face rings of simplicial cell complexes are analyzed and described, and a number of well-known results on the combinatorics of simplicial partitions are generalized. In particular, we give an explicit expression for the operator on f - and h-vectors that is defined by a barycentric subdivision, derive analogues of the Dehn­Sommerville relations for simplicial cellular decompositions of spheres and manifolds, and obtain a generalization of the well-known Stanley criterion for the existence of regular sequences in the face rings of simplicial cell complexes. As an application, a class of manifolds with a torus action is constructed, and generalizations of some of our previous results on the moment­angle complexes corresponding to triangulations are proved.

1. INTRODUCTION This pap er fo cuses on spaces with a sp ecial cellular decomp osition, simplicial cell complexes. In combinatorics, these complexes corresp ond to the so-called simplicial partially ordered sets. In top ology, simplicial cell complexes have b een studied virtually since the emergence of triangulations and combinatorial metho ds. However, in combinatorics, the systematic study of simplicial partially ordered sets started only around the mid-1980s (see [13]). Many fundamental constructions of commutative and homological algebra that were develop ed for studying triangulations are naturally carried over to simplicial cell complexes. For instance, in [14], an imp ortant concept of a face ring (or a Stanley­Reisner ring ) of a simplicial partially ordered set is intro duced and the main algebraic prop erties of these rings are describ ed. In [9], a group of projectivities of a triangulation is defined. The study of this group has led to the construction of an interesting class of simplicial cell complexes, called the unfoldings of triangulations. In the same work, on the basis of this construction, an explicit combinatorial description was obtained for the Hilden­Montesinos branched covering of an arbitrary closed oriented manifold over the 3-sphere. In the combinatorics of simplicial complexes, the so-called bistel lar subdivisions are used, which play an imp ortant role in applications to torus actions (see [2, Ch. 7]). In this way, one naturally encounters simplicial cellular decomp ositions b ecause the application of a bistellar sub division to a triangulation yields, in general, only a simplicial cellular decomp osition.
1 Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia.

E-mail: buchstab@mendeleevo.ru
2 Faculty of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119992 Russia.

E-mail: tpanov@mech.math.msu.su

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V.M. BUCHSTABER, T.E. PANOV

Simplicial cell complexes have found imp ortant applications in the study of torus manifolds (see [11]). A torus manifold is a 2n-dimensional smo oth compact closed manifold M on which an action of the n-dimensional compact torus T is defined which is effective and has at least one fixed p oint. Torus manifolds were intro duced in [8] as a far-reaching generalization of algebraic nonsingular compact toric varieties. The fixed p oints of the effective action of a half-dimensional torus lead to the rich combinatorial structure of the orbit space Q = M/T , which often allows one to describ e the torus manifold M itself. Under the additional condition that the action is local ly standard, the orbit space Q is a manifold with corners (a particular case of manifolds with corners is given by simple p olyhedra). The set of faces of a manifold with corners forms a simplicial partially ordered set with the order given by reverse inclusion. The study of manifolds with corners (or, equivalently, simplicial partially ordered sets) as orbit spaces of torus manifolds allows one to interpret many imp ortant top ological invariants of these manifolds in combinatorial terms. For example, as is shown in [11], equivariant cohomology of a torus manifold M is isomorphic to the face ring Z[Q] of the orbit space Q in a numb er of natural cases (for instance, when all faces in Q are acyclic). According to another result of [11], the cohomology of a torus manifold vanishes in o dd dimensions if and only if all faces of the orbit space Q are acyclic. Let us briefly describ e the contents of the pap er. In Section 2, we intro duce the basic concepts, simplicial and pseudosimplicial cell complexes, and describ e the relations b etween them and their prop erties. These cell complexes provide useful approximations to classical triangulations (simplicial complexes); in particular, the concept of barycentric sub division is intro duced for these complexes. The barycentric sub division of a pseudosimplicial cell complex is a simplicial cell complex, while the barycentric sub division of a simplicial cell complex is a simplicial complex (Prop osition 2.4). In Section 3, we consider f - and h-vectors of cell complexes. In the case of simplicial complexes, a transition to the barycentric sub division yields a linear op erator on the spaces of f - and h-vectors. We present an explicit expression for the matrix of this op erator (Lemmas 3.1 and 3.2). The op erator constructed can formally b e applied to the f -vector of any cell complex. The application of this op erator to the f -vector of a pseudosimplicial cell complex yields an f -vector of a simplicial cell complex, whereas its application to the f -vector of a simplicial cell complex yields an f -vector of a simplicial complex. The main result of this section is the derivation of analogues of the Dehn­ Sommerville relations for pseudosimplicial and simplicial cell complex of manifolds (Theorems 3.4 and 3.5 and Corollary 3.6). In Section 4, we intro duce the concept of branched combinatorial covering and prove that a space X is a simplicial cell complex if and only if there exists a branched combinatorial covering X K for a certain simplicial complex K (Theorem 4.1). In Section 5, we study the prop erties of the face rings of simplicial cell complexes intro duced by Stanley [14]. Here, we obtain results on the functoriality of such rings with resp ect to simplicial mappings (Prop ositions 5.3, 5.9, and 5.10) and present conditions under which these rings contain linear systems of parameters (Theorem 5.4 and Lemma 5.5); this is imp ortant for applications to torus actions. In Section 6, we intro duce and study a functor that assigns a space ZS with an action of the torus T m to any simplicial cell complex S of dimension n - 1 with m vertices; the space ZS generalizes the concept of a moment­angle complex ZK (see [6] and [2, Ch. 7]). Like for simplicial decomp ositions, ZS is a manifold for piecewise linear simplicial cellular decomp ositions of spheres (Theorem 6.3). The toral rank of a space X is the maximal numb er k such that there exists an almost free T k -action on X . We prove that, for any simplicial cell complex S of dimension n - 1 with m vertices, the toral rank of the space ZS is no less than m - n (Theorem 6.4). This result leads to an interesting relation b etween the well-known toral rank conjecture of Halp erin [7] and the combinatorics of triangulations (Corollary 6.5).
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2. SIMPLICIAL AND PSEUDOSIMPLICIAL CELL COMPLEXES An abstract simplicial complex on a set M is a collection K = { } of subsets in M such that, for every K , all subsets in (including ) also b elong to K . A subset K is called an (abstract) simplex of the complex K . One-element subsets are called vertices. The dimension of a simplex K is the numb er of its elements minus one: dim = | | - 1. The dimension of an abstract simplicial complex is the maximal dimension of its simplices. Along with abstract simplices, we consider geometric simplices, which are convex hulls of sets of affinely indep endent p oints in Rn . A geometric simplicial complex (or a polyhedron ) is a set P of geometric simplices of arbitrary dimensions lying in a certain Rn , such that every face of a simplex from P lies in P and the intersection of any two simplices from P is a face of each of them. A p olyhedron P is called a geometric realization of an abstract simplicial complex K if there exists a one-to-one corresp ondence b etween the vertex sets of the complex K and the p olyhedron P under which simplices of the complex K are mapp ed into the vertex sets of the simplices of the p olyhedron P . For every simplicial complex K , there exists a unique, up to a simplicial isomorphism, geometric realization, which is denoted by |K |. In what follows, we will not distinguish b etween abstract simplicial complexes and their geometric realizations. Let S b e an arbitrary partially ordered set. Its order complex ord(S ) is the set of all chains x1 < x2 < ... < xk , xi S . It is obvious that ord(S ) is a simplicial complex. The barycentric subdivision K of a simplicial complex K is defined as the order complex ord(K \ ) of the partially ordered (with resp ect to inclusion) set of nonempty simplices of the complex K . A partially ordered set S is called simplicial if it contains the least element 0 and, for any S , the lower segment 0, = S : 0 is the partially ordered (with resp ect to inclusion) set of faces of a certain simplex. Intro duce the rank of elements of the set S putting rk 0 = 0 and rk = k if 0, is identified with the set of faces of a (k - 1)-simplex. We also put dim S = maxS rk - 1. We call elements of rank 1 the vertices of the set S . Then, any element of rank k contains exactly k vertices. It is obvious that the set of all simplices of a certain simplicial complex is a simplicial partially ordered set with resp ect to inclusion. However, not all simplicial partially ordered sets are obtained in this manner. Recall that a cel l complex is a Hausdorff top ological space X represented as the union eq of i pairwise disjoint sets eq (called cel ls ) such that, for every cell eq , a unique mapping of a q -dimensional i i closed ball Dq into X is fixed (the characteristic mapping ) whose restriction to the interior of the ball D q is a homeomorphism onto eq . (Here, as usual, we assume that the interior of a p oint is the i p oint itself.) Moreover, it is assumed that the following axioms hold: (C) The b oundary of a cell eq is contained in the union of a finite numb er of cells er of dimension j i r < q. (W) A subset Y X is closed if and only if, for any cell eq , the intersection Y eiq is closed. i In what follows, we will always assume that partially ordered sets and simplicial and cell complexes are finite. Let S b e a simplicial partially ordered set. To each element S \ 0, we assign a simplex with the set of faces given by 0, and glue together all these geometric simplices according to the order relation in S . Then, we obtain a cell complex in which the closure of each cell is identified with a simplex, the structure of faces b eing preserved; moreover, all characteristic mappings are emb eddings. This complex is called a simplicial cel l complex and is denoted by |S |. When S is a set of simplices of a certain simplicial complex K , the space |S | coincides with the geometric realization
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V.M. BUCHSTABER, T.E. PANOV '$ '$ '$ s s s s s s &% &% &%

(a)

(b)

(c)

Fig. 1. Pseudosimplicial cellular decompositions of a circle

of |K |. Henceforth, we will not distinguish b etween simplicial partially ordered sets and simplicial cell complexes. Example 2.1. A cell complex obtained by identifying two (n - 1)-dimensional simplices along their b oundaries (with the structure of faces preserved) is a simplicial cell complex. The corresp onding simplicial partially ordered set is not a set of faces of any simplicial complex for n > 1. This example is a particular case of the general construction of a union of simplicial cell complexes along a common sub complex. Note that simplicial cell complexes represent a minimal extension of the class of simplicial complexes that is closed with resp ect to the op eration of taking a union along a common sub complex. Recall that a continuous mapping of cell complexes is called cel lular if the image of any cell lies in a union of cells of the same or lower dimension. A cellular mapping : S1 S2 of simplicial cell complexes is called simplicial if the image of every simplex from S1 is a simplex in S2 . It is clear that a simplicial mapping is a mapping of the corresp onding partially ordered sets (i.e., preserves order). A simplicial mapping is called a simplicial isomorphism if there exists a simplicial inverse for it. In geometrical terms, one may assume that a simplicial mapping of simplicial complexes (p olyhedra) is linear on simplices. By analogy with simplicial complexes, we define the barycentric subdivision S of a simplicial cell complex S as the order complex ord S \ 0 . It follows directly from the definition that the barycentric sub division S is a simplicial complex. Thus, a simplicial mapping b etween arbitrary simplicial cell complexes can b e assumed to b e linear on the simplices of the barycentric sub division. Remark. The identity mapping of a simplicial cell complex defines a cellular, but not simplicial, mapping S S such that its inverse is not a cellular mapping. A cell complex X is called pseudosimplicial if the characteristic mapping of any cell is a mapping of the simplex q into X whose restriction onto each face is the characteristic mapping for a certain other cell. Example 2.2. Figure 1 shows three pseudosimplicial cellular decomp ositions of a circle. The first of them (a) is not a simplicial cellular decomp osition. The second (b) is a simplicial cellular decomp osition but is not a simplicial complex. The third (c) is a simplicial complex. Lemma 2.3. A pseudosimplicial cel l complex X is a simplicial cel l complex if and only of the characteristic mapping of any of its cel ls is an embedding. Pro of. For a given pseudosimplicial cell complex X , we intro duce a partially ordered set S whose elements are closures of cells and the order relation corresp onds to emb edding. Let all the characteristic mappings b e emb eddings. Then, it is obvious that, for every cell eq , the corresp onding i characteristic mapping is a homeomorphism of q onto eiq , and the characteristic mapping of any cell ep eiq is a restriction of this homeomorphism onto a certain p-face of the simplex q . Therefore, j S is a simplicial partially ordered set and X is a simplicial cell complex |S |. The converse assertion is obvious. Pseudosimplicial cellular decomp ositions of manifolds play an imp ortant role in various constructions of the low-dimensional top ology, where they are referred to as ideal or singular triangulations (see, e.g., [12]).
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Let X b e a pseudosimplicial cell complex. Its barycentric subdivision is a cell complex X whose cells are images of op en simplices of the standard barycentric sub division of simplices q under all p ossible characteristic mappings q X of the cells of the complex X . The correctness of this definition follows immediately from the definition of a pseudosimplicial cell complex. Remark. The ab ove "geometrical" definition of a barycentric sub division is consistent with the "combinatorial" definitions, intro duced ab ove, of the barycentric sub division of a simplicial complex and a simplicial partially ordered set. However, simple examples show that, in the case of an arbitrary pseudosimplicial cell complex X , the complex X differs from the order complex of the partially ordered set of the closures of the cells of the complex X . It follows directly from the definition that X is a pseudosimplicial cell complex. Indeed, the following more general prop osition holds (which is a part of mathematical folklore; see, e.g., [12]). Prop osition 2.4. The barycentric subdivision X of a pseudosimplicial cel l complex X is a simplicial cel l complex, while the second barycentric subdivision X is a simplicial complex. Pro of. By Lemma 2.3, to prove the first assertion, it suffices to verify that all characteristic mappings of the cells of the complex X are emb eddings. Supp ose the contrary, i.e., some two p oints x and y go to a single p oint under the characteristic mapping of a certain q -dimensional cell eq of the complex X . One may assume that q is the minimal dimension of such cells. By the definition of X , the characteristic mapping of its cell eq is a restriction of the characteristic mapping q X of a certain cell of the complex X onto a certain q -dimensional simplex of the barycentric sub division of the standard simplex q . Since q is minimal, at least one of the p oints x or y is contained in the interior of the simplex q . Now, using the fact that the restriction of a characteristic mapping onto the interior of a cell is one-to-one, we arrive at a contradiction. This proves the first assertion, which implies the second. 3. f -VECTORS AND THE DEHN­SOMMERVILLE RELATIONS Let X b e a cell complex of dimension n - 1. Denote by fi the numb er of its i-dimensional cells. An integer vector f (X ) = (f0 ,... ,fn-1 ) is called the f -vector of the complex X . It is convenient to assume that f-1 = 1. Intro duce the h-vector of the complex X as an integer vector (h0 ,h1 ,... ,hn ) determined from the equation h0 tn + ... + hn
-1

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

-1

+ ... + f

n-1

.

(3.1)

Note that the f -vector and the h-vector carry the same information ab out the cell complex and are expressed in terms of each other by linear relations, namely,
k

hk =
i=0

(-1)k

-i

n-i f n-k

n i-1

,

f

n-1-k

=
q =k

q h kn

-q

,

k = 0,... ,n.

(3.2)

In particular, h0 = 1 and hn = (-1)n (1 - f0 + f1 + ... + (-1)n fn-1) = (-1)n (1 - (X )), where (X ) is the Euler characteristic of the complex X . First, we describ e some prop erties of f -vectors in the case when X = S is a simplicial cell complex (i.e., the closures of cells form a simplicial partially ordered set). The join of simplicial partially ordered sets S1 and S2 is a set S1 S2 consisting of elements 1 2 , where 1 S1 and 2 S2 . A partial order relation is intro duced as follows: 1 2 1 2 if 1 1 and 2 2 . It is clear that S1 S2 is a simplicial partially ordered set. The space (cell complex) |S1 S2 | is known in top ology as the join of the spaces |S1 | and |S2 |. A simplicial cell complex S is called pure if all of its maximal simplices have the same dimension. Consider two pure complexes S1 and S2 of the same dimension, select maximal simplices 1 S1
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and 2 S2 , and fix a certain identification for them (this is done, for example, by cho osing the order of vertices in 1 and 2 ). A simplicial cell complex S1 #1 ,2 S2 obtained by gluing together S1 and S2 along 1 and 2 followed by the removal of the simplex = 1 = 2 is called the connected sum of the complexes S1 and S2 . When the result is indep endent of the choice of the simplices 1 and 2 and the metho d of their identification, we will use an abbreviated notation S1 # S2 . Let us express the f -vector and the h-vector of the connected sum S1 # S2 in terms of the f -vectors and h-vectors of the complexes S1 and S2 . Let dim S1 = dim S2 = n - 1; then, we have fi (S1 # S2 ) = fi (S1 )+ fi (S2 ) - f
n-1

n , i +1

i = 0, 1,... ,n - 2,

(S1 # S2 ) = f

n-1

(S1 )+ f

n-1

(S2 ) - 2.

Then, (3.2) implies that h0 (S1 # S2 ) = 1, hi (S1 # S2 ) = hi (S1 )+ hi (S2 ), hn (S1 # S2 ) = hn (S1 )+ hn (S2 ) - 1. Now, let dim S1 = n1 - 1 and dim S2 = n2 - 1; then, we have the following equation for the join S1 S2 :
n1 -1

i = 1, 2,... ,n - 1,

(3.3)

fk (S1 S2 ) =
i=-1

fi (S1 )f

k -i-1

(S2 ),

k = -1, 0,... ,n1 + n2 - 1.

Put h(S ; t) = h0 + h1 t + ... + hn tn . Then, the preceding formula and (3.1) imply h(S1 S2 ; t) = h(S1 ; t)h(S2 ; t). (3.4)

Next, we will need transformation formulas for the f - and h-vectors of simplicial cell complexes under barycentric sub divisions. Intro duce the matrix
i

B = (bij ),

0 i, j n - 1,

bij =
k =0

(-1)k

i +1 (i - k +1)j k

+1

.

One can verify that bij = 0 for i > j (i.e., B is an upp er triangular matrix) and bii = (i +1)!. Thus, the matrix B is invertible. Lemma 3.1. Let S be the barycentric subdivision of an (n - 1)-dimensional simplicial cel l complex S . Then, the f -vectors of the complexes S and S are related by the formula
n-1

fi (S ) =
j =i

bij fj (S ),

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

i.e., f (S ) = B f (S ). Pro of. Consider the barycentric sub division of a j -dimensional simplex j and supp ose that bij - is the numb er of i-simplices in (j ) that do not lie in j . Then, we have fi (K ) = n=i1 bij fj (K ). j Let us prove that bij = bij . Indeed, it is obvious that the numb er bij satisfies the following recurrent
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relation: bij = (j +1)bi-
1,j -1

+

j +1 bi- 2

1,j -2

+ ... +

j +1 b j - i +1

i-1,i-1

.

Hence, one can easily derive by induction that bij is defined by the same formula as bij . Now, let us intro duce the matrix
p

D = (dpq ),

0 p, q n,

dpq =
k =0

(-1)k

n +1 (p - k)q (p - k +1)n k

-q

(here, we assume that 00 = 1). Lemma 3.2. The h-vectors of the complexes S and S are related by
n

hp (S ) =
q =0

dpq hq (S ),

p = 0,... ,n,

i.e., h(S ) = D h(S ). Moreover, the matrix D is invertible. Pro of. The lemma is proved by a routine verification with the use of Lemma 3.1, formulas (3.1), and a numb er of identities for binomial co efficients, which can b e found, for example, in [3]. If we add the comp onent f-1 = 1 to the f -vector and appropriately change the matrix B , then we obtain the relation D = C -1 BC , where C is the transition matrix from the h-vector to the f -vector (the explicit form of this matrix can easily b e obtained from (3.1)). This implies the invertibility of the matrix D. Thus, the barycentric sub division induces invertible linear op erators B and D on the f - and h-vectors of simplicial cell complexes. The following fact is a classical result of combinatorial geometry: if a simplicial complex K is a triangulation of an (n - 1)-dimensional sphere, i.e., |K | S n-1 , then its h-vector is symmetric: = hi = hn-i , i = 0,... ,n. (3.5)

These equations (as well as their different expressions in terms of f -vectors) are known as the Dehn­Sommervil le relations. Note that h0 = 1 for any simplicial complex and the first relation h0 = hn is equivalent to the formula for the Euler characteristic (see (3.1)). Various pro ofs of the Dehn­Sommerville relations, as well as an account of the history of the question, can b e found in [2]. We will need the following result, which is due to Klee [10]. Prop osition 3.3. The Dehn­Sommervil le relations are the most general linear equations that hold true for the f -vectors of al l triangulations of spheres. Pro of. In [10], this prop osition was proved in terms of f -vectors. However, the use of h-vectors significantly simplifies the pro of. It is sufficient to show that the affine hull of the h-vectors (h0 ,h1 ,... ,hn ) of the triangulations of spheres is an n -dimensional plane (recall that h0 = 1 2 always). This can b e done, for example, by indicating n +1 triangulations of spheres with affinely 2 indep endent h-vectors. Put Kj := j n-j , j = 0, 1,... , n , where j stands for the 2 b oundary of a j -dimensional simplex. Since h( j ) = 1 + t + ... + tj , it follows from (3.4) that h(Kj ) = 1 - tj +1 1 - tn-j 1-t 1-t
+1

.
n 2

Thus, h(Kj +1 ) - h(Kj ) = tj +1 + higher order terms, j = 0, 1,... , j = 0, 1,... , n , are affinely indep endent. 2
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Theorem 3.4. If S is a simplicial cel lular decomposition of an (n - 1)-dimensional sphere, then the h-vector h(S ) satisfies the Dehn­Sommervil le relations (3.5). Pro of. Consider the barycentric sub division S . By Lemma 3.2, h(S ) = D h(S ), where the vector h(S ) is symmetric b ecause S is a triangulation of a sphere. A routine test with the use of well-known binomial identities shows that the matrix D (and its inverse) transforms symmetric vectors into symmetric vectors (this is equivalent to dpq = dn+1-p,n+1-q , i.e., the matrix D is centrally symmetric). However, this can b e proved without resort to computations and even without using the explicit form of the matrix D from Lemma 3.2. Indeed, the Dehn­Sommerville relations determine a linear subspace W of dimension k = n + 1 (or an affine space of dimension n if 2 2 we add the relation h0 = 1) in the space Rn+1 (with co ordinates h0 ,... ,hn ). We have to verify that this subspace is invariant with resp ect to the invertible linear op erator D. To this end, it suffices to cho ose an arbitrary basis e1 ,... ,ek in W and verify that Dei W for all i. However, the subspace W admits a basis comp osed of the h-vectors of simplicial spheres (see the pro of of Prop osition 3.3). Since the barycentric sub division of a simplicial sphere is again a simplicial sphere, the images Dei , i = 1,... ,k, also satisfy the Dehn­Sommerville relations. Hence, the subspace W is D -invariant. This implies that the vector h(S ) = D-1 h(S ) satisfies the Dehn­Sommerville relations. Remark. The pro of of the Dehn­Sommerville relations for Eulerian partial ly ordered sets (which include simplicial cellular decomp ositions of spheres as a particular case) was first obtained by Stanley [13, (3.40)]. In [2], a generalization of the Dehn­Sommerville relations for the triangulations of arbitrary top ological manifolds was obtained. Namely, the differences b etween symmetric comp onents of the h-vector of the triangulation of an (n - 1)-dimensional manifold M are expressed in terms of its Euler characteristic: hn
-i

- hi = (-1)i (M ) - (S

n-1

)

n n = (-1)i (hn - 1) , i i

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

(3.6)

In particular, if M = S n-1 or n is o dd, we obtain relations (3.5). Arguments analogous to those used in the pro of of Theorem 3.4 allow us to generalize this result to arbitrary simplicial cellular decomp ositions of manifolds. Theorem 3.5. Let S be a simplicial cel lular decomposition of an (n - 1)-dimensional manifold M . Then, the h-vector h(S ) = (h0 ,... ,hn ) satisfies relations (3.6). Pro of. Let S b e the barycentric sub division of the complex S and D b e the matrix (op erator) from Lemma 3.2 such that h(S ) = D h(S ). Let, next, A denote an affine subspace of dimension k = n in Rn+1 (with co ordinates h0 ,... ,hn ) defined by the relations from the text of the theorem. 2 Since S is a simplicial complex, just as in the pro of of Theorem 3.4, we have to verify that A is invariant with resp ect to D. To this end, it suffices to cho ose a basis in A (i.e., a set of n +1 affinely 2 indep endent vectors) that consists of the h-vectors of simplicial decomp ositions of the manifold M . This can b e done as follows. Consider the h-vectors of n +1 triangulations of an (n - 1)-sphere 2 of the form j n-j , which constitute a basis in the subspace W sp ecified by the relations hi = hn-i (see Prop osition 3.3). Then, consider the connected sums K # ( j n-j ), where K is a certain fixed triangulation of the manifold M . Then, the corresp onding h-vectors form a basis in A (this easily follows from relations (3.3)). Corollary 3.6. Analogues of Theorems 3.4 and 3.5 hold for pseudosimplicial cel lular decompositions of spheres and manifolds, respectively. Pro of. The pro of follows from Prop osition 2.4.
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&% &%

(a)

(b)
2

Fig. 2. Cellular decompositions of the disk D

Example 3.7. In Fig. 2, two cellular decomp ositions of the disk D2 are shown. The first of them (a) has two 0-dimensional, two 1-dimensional, and one 2-dimensional cells and is pseudosimplicial. This can b e seen as follows. Let us realize D2 as a unit disk in C and the simplex 2 as a quarter of the unit disk that is sp ecified by the conditions Re z 0 and Im z 0. Then, the mapping z z 4 defines a characteristic mapping for the two-dimensional cell. The cellular decomp osition of a disk shown in Fig. 2b is not pseudosimplicial. Indeed, otherwise, gluing together two such disks along the b oundary, we would obtain a pseudosimplicial cellular decomp osition of the two-dimensional sphere. However, the f -vector of this cellular decomp osition is equal to (1, 1, 2), and therefore the corresp onding h-vector (1, -2, 2, 1) do es not satisfy the second of the Dehn­Sommerville relations h1 = h2 . The first relation h0 = h3 , which is equivalent to the formula for the Euler characteristic, is obviously valid. The definition of a pseudosimplicial cellular decomp osition implies that this decomp osition must contain cells of all dimensions. The decomp osition shown in Fig. 2b gives an example of a situation where the converse prop osition do es not hold. 4. BRANCHED COMBINATORIAL COVERINGS Here, we characterize simplicial cell complexes as a sp ecial class of branched coverings over simplicial complexes. In this section, we deal with geometric simplicial complexes (i.e., with p oly hedra) K . An open simplex K is defined as the relative interior of a certain simplex K . When is a vertex, we put = . Let X b e a compact Hausdorff top ological space and K b e a simplicial complex. A continuous mapping p : X K is called a branched combinatorial covering over K if the following two conditions are fulfilled: (1) for any op en simplex K , the preimage p numb er of op en sets Ui ( ): p
-1 -1

is a nonempty disjoint union of a finite i = 1,... ,I ( );

( ) =
i

Ui ( ),

(2) the mapping p : Ui ( ) is a homeomorphism for any i. It follows immediately from the definition that the op en sets Ui ( ) corresp onding to all simplices K and all i define a cellular decomp osition of the space X . Theorem 4.1. A space X is a simplicial cel l complex if and only if there exists a branched combinatorial covering of X K for a certain simplicial complex K . Pro of. Let X b e a simplicial cell complex with vertex set M. Intro duce a simplicial complex KS on the vertex set M as follows. We say that a subset M is a simplex in KS if there exists a simplex X with the vertex set . Then, the mapping X KS , which is the identity mapping on the vertex set M, sends the simplex to the corresp onding simplex and is a branched combinatorial covering by construction. Now, let K b e a simplicial complex and p : X K b e a certain branched combinatorial covering. Then, X is a cell complex with cells of the form Ui ( ), K . In order to establish that X is a
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simplicial cell complex, it is necessary to verify that the set of cells in the closure of each cell Ui ( ) forms a partially ordered (by inclusion) set of faces of a certain simplex. This is equivalent to the fact that the restriction of the pro jection p onto the closure of each cell is a homeomorphism (since the corresp onding prop erty holds for K ). Put = U i ( ). We have to prove that p : is a homeomorphism. Since and are compact and Hausdorff, it is sufficient to prove that p : is one-to-one. Obviously, this mapping is epimorphic; supp ose that p(x1 ) = p(x2 ) = y for some two different p oints x1 ,x2 . Let us connect the p oints x1 and x2 by a path segment : [0, 1] X such that (0) = x1 , (1) = x2 , and (s) Ui ( ) for 0 < s < 1. Then, p : [0, 1] K is a lo op with the b eginning and end at y , whose interior lies in . Since is a simplex, there exists a contracting homotopy F : [0, 1] â [0, 1] K such that F (s, 1) = p (s), F (s, 0) = y , and F (s, t) for 0 < s < 1 and 0 < t < 1. Put i = p
-1

F [0, 1], 1 i



(note that 1 = ). Each subset i X is connected (as the closure of the connected subset p-1 (F ((0, 1), 1 ))) and compact as the preimage of a compact subset under a prop er mapping. i Consider the upp er limit =
N iN

i .

Since each subset i is connected and compact, is also connected and compact. However, on the other hand, p() = y , so that consists of a finite numb er of p oints by the definition of a branched combinatorial covering. The contradiction obtained completes the pro of. 5. FACE RINGS The face ring of a simplicial complex is an imp ortant concept that allows one to translate combinatorial prop erties into the language of commutative and homological algebra. This ring was intro duced by Stanley and Reisner (see [15]) and is often called a Stanley­Reisner ring. A generalization of the face ring to arbitrary simplicial partially ordered sets was first intro duces in [14], where imp ortant algebraic prop erties of these rings were also describ ed. However, in the present account, which is intended for top ological applications, we follow [11] (see also [2, Ch. 4]). First, supp ose that K is a simplicial complex and identify its vertex set M with the index set [m] = {1,... ,m}. Let Z[v1 ,... ,vm ] b e a graded ring of p olynomials with integer co efficients and with m degree-2 generators. The Stanley­Reisner ring (or the face ring ) of a simplicial complex K is the graded quotient ring Z[K ] = Z[v1 ,... ,vm ]/IK , where IK is the homogeneous ideal generated by the monomials vi1 · ... · vik for which {i1 ,... ,ik } is not a simplex in K . Next, we slightly mo dify the previous definition by extending b oth the set of generators of the p olynomial ring and the set of relations. In this case, the quotient ring Z[K ] remains unchanged, and the new definition thus obtained can easily b e generalized to arbitrary simplicial partially ordered sets. For any two simplices , K , denote by their unique greatest lower b ound (i.e., the maximal simplex that is contained simultaneously in and ). The greatest lower b ound always exists but may b e an empty simplex K . On the other hand, the least upp er b ound of the simplices and (i.e., the minimal simplex that simultaneously contains and ) may not exist. If the least upp er b ound exists, it is unique, and we denote it by .
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Consider a p olynomial ring Z[v : K \ ], which has one generator p er each nonempty simplex in K . Intro duce a grading by putting deg v = 2| |. In addition, identify v with 1. The following prop osition gives an alternative "less economic" representation for the face ring Z[K ]. Prop osition 5.1. There is a canonical isomorphism of graded rings Z[v : K \ ]/I Z[K ], = where I is the ideal generated by al l elements of the form v v - v
v

.

Here, we assume that v = 0 if the least upper bound does not exist. Pro of. The isomorphism is established by a mapping that sends v into i vi . Now, let S b e an arbitrary simplicial cell complex. Then, for any two elements , S , the set of their least upp er b ounds may consist of more than one element (but may also b e empty). The set of greatest lower b ounds is always nonempty; moreover, it consists of a single element provided that = . Intro duce a graded p olynomial ring Z v : S \ 0 , where deg v = 2 rk . We also formally set v0 = 1. The face ring of a simplicial cell complex S is the quotient ring Z[S ] := Z v : S \ 0 /IS , where IS is the ideal generated by all elements of the form v v - v


v .

(5.1)

In particular, if = , then v v = 0 in Z[S ]. The relations that generate the ideal IS allow one to express the pro duct of the generators v and v that corresp ond to incomparable elements , S as a sum of monomials each of which is a pro duct of ordered generators. In the algebraic literature, such relations are called straightening relations. Example 5.2. Consider the complex S describ ed in Example 2.1 for n = 2. Thus, S is obtained by gluing together two segments along their b oundaries. We have two elements of rank 1 (two vertices), say, 1 and 2 , and two elements of rank 2 (two maximal simplices), say, 1 and 2 . Then, the face ring is expressed as Z[S ] = Z[v1 ,v2 ,v1 ,v2 ]/(v1 + v2 = v1 v2 , v1 v2 = 0), where deg v1 = deg v2 = 2 and deg v1 = deg v2 = 4. It was p ointed out in [2, Ch. 3] that a simplicial mapping : K1 K2 of simplicial complexes induces a mapping of the face rings : Z[K2 ] Z[K1 ], i.e., the face ring has a contravariant character. A similar prop erty also holds for arbitrary simplicial cell complexes. Prop osition 5.3. Let : S1 S2 be a simplicial mapping. Define a mapping of the graded polynomial rings : Z v : S2 \ 0 Z v : S1 \ 0 , v v ,

where the sum is over al l simplices S1 such that ( ) = and dim = dim . Then, the mapping induces a mapping of the face rings Z[S2 ] Z[S1 ] (which wil l be denoted by the same symbol ). Pro of. One can directly verify that (IS2 ) IS1 .
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Let S b e a simplicial cell complex of dimension n. By analogy with the ab ove pro cedure, we define a face ring k[S ] over an arbitrary commutative ring k with unity (along with k = Z, we are interested in the case k = Q). A sequence t1 ,... ,tn of algebraically indep endent homogeneous elements of the ring k[S ] is called a homogeneous system of parameters if k[S ] is a finitely generated k[t1 ,... ,tn ]-mo dule. Thus, the sequence of homogeneous elements t1 ,... ,tk of length k n is a part of the homogeneous system of parameters if and only if dim k[S ]/(t1 ,... ,tk ) = n - k, where dim denotes the Krul l dimension. A system of parameters that consists of degree-2 elements is called linear. A sequence of elements t1 ,... ,tk k[S ] is called regular if k[S ] is a free k[t1 ,... ,tk ]mo dule. A homogeneous regular sequence is a part of the homogeneous system of parameters; however, the converse is not true. A ring k[S ] is called a Cohen­Macaulay ring (and a complex S is called a Cohen­Macaulay complex ) if it admits a regular sequence of length n = dim k[S ] = dim S +1. An example of the Cohen­Macaulay ring is given by a simplicial cellular decomp osition of a sphere [14]. For every simplex S , define the restriction homomorphism s : k[S ] k[S ]/(v : ). Let dim = k - 1 and {i1 ,... ,ik } b e the vertex set of the simplex . Then, it is obvious that the image of the homomorphism s is a p olynomial ring k[vi1 ,... ,vik ] of k degree-2 generators. The following prop osition gives a characterization of linear systems of parameters in the face rings of simplicial cell complexes and generalizes the corresp onding prop osition [15, Lemma I I I.2.4] for simplicial complexes (see also [6, Theorem 7.2]). Theorem 5.4. A sequence t = (t1 ,... ,tn ) of degree-2 elements of the ring k[S ] is a linear system of parameters if and only if, for every simplex S , the sequence s (t) generates the ring of polynomials k[vi : i ]. Pro of. First, supp ose that t is a linear system of parameters. The mapping s induces an epimorphism of quotient rings: k[S ]/(t) k[vi : i ]/s (t). Since t is a system of parameters, it follows that dim k[S ]/(t) = 0, i.e., dimk k[S ]/(t) < . Therefore, k[vi : i ]/s (t) < . However, this happ ens only when s (t) multiplicatively generates a p olynomial ring over k. Now, supp ose that, for every S , the set s (t) generates the ring k[vi : i ]. Then, we have dimk
S

k[vi : i ]/s (t) < .

Moreover, the sum s : k[S ] S k[vi : i ] of restriction homomorphisms is a monomorphism [2, Theorem 4.8]. Hence, dimk k[S ]/(t) < as well (see [5, Lemma 4.7.1]). Thus, t is a linear system of parameters. It is clear that the theorem remains valid if we consider only restrictions s (t) onto maximal simplices S . In particular, if S is a pure complex (all maximal simplices have dimension n - 1), then a sequence t1 ,... ,tn is a linear system of parameters if and only if its restriction onto each (n - 1)-simplex yields a basis in the space of linear forms (over k). Let KS b e the simplicial complex constructed by the simplicial cell complex S in the pro of of Theorem 4.1. It is obvious that the ring k[KS ] coincides with a subring of the ring k[S ] generated by degree-2 elements. Lemma 5.5. The ring Q[S ] admits a linear system of parameters.
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Pro of. If S is a simplicial complex, then the ring Q[S ] is generated by linear elements, and the assertion of the lemma follows from the No ether normalization lemma (see, e.g., [5, Theorem 1.5.17]). In the general case, Theorem 5.4 implies that the linear system of parameters in the ring Q[KS ] is also a linear system of parameters for Q[S ]. Example 5.6. In the face ring from Example 5.2, the set v1 ,v2 forms a linear system of parameters, while the elements v1 and v2 are the ro ots of the algebraic equation x2 - (v1 v2 )x = 0. The question of the existence of a liner system of parameters in the ring Z[S ] is much more delicate even when S is a simplicial complex (in this case, the No ether normalization lemma states only the existence of a nonlinear homogeneous system of parameters). This question is closely related to the calculation of the rank of a freely acting torus on certain manifolds (see Section 6 b elow). Here, we only note that an example of a simplicial complex K for which the ring Z[K ] do es not admit a linear system of parameters is given by the b oundary of a cyclic polyhedron C n (m) with m 2n 16 vertices. This example was constructed in [6, Example 1.22] (it is also presented in [2, Example 6.33]). Lemma 5.7. Let v1 ,... ,vm be degree-2 elements in the ring Z[S ] that correspond to the vertices of the complex S . Then, the fol lowing identity holds in the ring Z[S ]: (1 + v1 ) · ... · (1 + vm ) =
S

v .

(5.2)

Pro of. Let ver denote the vertex set of the element S . Relations (5.1) in the ring Z[S ] imply the relations vi1 · ... · vik =
: ver ={i1 ,...,ik }

v .

(5.3)

Summing up these relations over all S , we obtain the required identity. The ring Z[S ] admits a canonical Nm -grading defined as mdeg v = 2 iver ei , where ei Nm is the ith basis vector. In particular, mdeg vi = 2ei . If all the rings are assumed to b e multigraded, then formula (5.2) is equivalent to the set of relations (5.3). It follows from Prop osition 5.1 that relations (5.3) generate the ideal IS in the case when S is a simplicial complex. However, the following example shows that, for an arbitrary simplicial partially ordered set S , the face ring Z[S ] may not b e isomorphic to the quotient ring of the Nm -graded p olynomial ring Z[v : S ] by relation (5.2). Example 5.8. Consider a simplicial cell complex S obtained by identifying two 2-simplices 1 and 2 at three their vertices. Consider two edges 1 1 and 2 2 that have common vertices, and let b e the opp osite vertex of the complex S . Then, the following relations hold in the ring Z[S ]: v1 v = v1 and v2 v = v2 ; however, relation (5.2) only implies v1 v + v2 v = v1 + v2 . The p olynomial PS = S v was considered by Alexander in [4] (1930) in relation to the formalization of the concept of simplicial complex in combinatorial top ology (however, as we p ointed out in the intro duction, the concept of face ring app eared much later). A natural question arises: To what extent a simplicial partially ordered set S is determined by its face ring Z[S ]? In the case of simplicial complexes, the following simple prop osition holds. Prop osition 5.9. The ring homomorphism F : Z[K2 ] Z[K1 ] is a homogeneous degree-0 isomorphism of Nm -graded face rings of two simplicial complexes if and only if F is induced by a simplicial isomorphism K1 K2 . Pro of. Let F b e a homogeneous degree-0 isomorphism of Nm -graded rings. Then, it establishes a bijection b etween the vertex sets of the complexes K1 and K2 . Therefore, F (PK1 ) = PK2 , and the assertion follows from relation (5.2). The converse is obvious.
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Even the simplest example of a simplex (when the face ring is a p olynomial ring) shows that one cannot replace "isomorphism of Nm -graded rings" by "isomorphism of graded rings" in the ab ove prop osition. In order that the isomorphism F : Z[K2 ] Z[K1 ] of graded rings b e induced by a simplicial isomorphism, one should additionally require that F (PK1 ) = PK2 . At the same time, we do not know whether there exist nonisomorphic simplicial partially ordered sets with isomorphic face rings. A simplicial mapping is called nondegenerate if its restriction onto each simplex is a simplicial isomorphism. The following prop osition is obvious. Prop osition 5.10. If : S1 S2 is a nondegenerate simplicial mapping, then we have (PS2 ) = PS1 .

Simple examples show that the relation (PS2 ) = PS1 do es not hold for arbitrary simplicial mappings. 6. TORUS ACTIONS Let K b e a simplicial complex with m vertices. In [2, § 5.2], we describ ed a cubic decomp osition cc(K ) of a cone cone K over the barycentric sub division of the simplicial complex K . This cubic decomp osition is obtained by identifying the cone cone(m-1 ) over the barycentric sub division of an (m - 1)-dimensional simplex with the standard triangulation of a cub e I m and considering the emb edding cone K cone(m-1 ) . (This emb edding is induced by the canonical emb edding K m-1 .) Next, in [2, § 7.2], we constructed a moment­angle complex ZK as the pullback that closes the commutative diagram ZK - - (D2 )m -- I
m

cc(K ) - - --

where the right arrow is a pro jection onto the orbit space of the standard action of an m-dimensional torus on a p olydisk. Thus, if dim K = n - 1, then ZK is an (m + n)-dimensional space with a T m -action. This space was first intro duced by Davis and Januszkiewicz in [6] in relation to the construction of top ological analogues of algebraic toric varieties. In [2], we devoted Chapters 7 and 8 to the study of various top ological prop erties of ZK and their relations to the combinatorics of the complex K . For instance, it is easy to prove that if K is a triangulation of an (n - 1)-dimensional sphere, then ZK is an (m + n)-dimensional manifold (see, for example, [2, Lemma 7.13]). Further in the present pap er, using Theorem 4.1, we generalize some of the constructions related to the space ZK to the case of arbitrary simplicial cell complexes. Let S b e an arbitrary simplicial cell complex with m vertices. Consider the comp osition cone S cone K I m of the mapping induced by the branched combinatorial covering p : S K from Theorem 4.1 and the ab ove-describ ed emb edding into a cub e. Define a space ZS with a T m -action that is induced by this comp osition: ZS -- -- ZK - - (D2 )m -- I
m

-- -- cone S - - cone K - -

Example 6.1. Let S b e a simplicial cell complex obtained by the identification of two (n - 1)dimensional simplices along their b oundaries (see Example 2.1). Then, ZS is obtained by the identification of two p olydisks (D2 )n along their b oundaries. Hence, ZS S 2n . In another asp ect, =
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this manifold with a T n -action was considered in [11] as the first example of a torus manifold that is not a quasitoric variety in the sense of [6]. The following prop osition follows obviously from the construction of the space ZS . Prop osition 6.2. The isotropy subgroups (stabilizers ) of the T m -action on ZS are coordinate subtori of the form T ver T m , where is a certain simplex in S . Let X be a space and K1 and K2 b e two triangulations of X , i.e., two simplicial complexes such that |K1 | |K2 | X . Two such triangulations are called combinatorial ly equivalent (or piecewise = = linearly isomorphic ) if there exists a simplicial complex K that is a sub division of b oth complexes K1 and K2 . A triangulation of an (n - 1)-dimensional sphere S n-1 is called piecewise linear (or, shortly, a PL-sphere ) if it is combinatorially equivalent to the b oundary of the simplex n . The concepts of combinatorial equivalence and PL-sphere are directly carried over to the simplicial cell complexes S by passing to the barycentric sub division S . Theorem 6.3. Let S be a piecewise linear simplicial cel lular decomposition of a sphere S with m vertices. Then, ZS is an (m + n)-dimensional manifold.
n-1

Pro of. Consider the complex dual to the simplicial complex S . This complex has one hyp erface (a face of co dimension 1) Fv p er each vertex v of the complex S , where Fv := starS v = { S : v S }. Faces of co dimension k are defined as nonempty intersections of the sets of k hyp erfaces. Since S is a PL-sphere, each i-dimensional face is piecewise linearly homeomorphic to an i-dimensional ball. For each vertex v S , denote by Uv an op en subset in cone(S ) obtained by removing from S all the faces that do not contain v . Then, {Uv } is an op en covering of the space cone(S ); moreover, each Uv is homeomorphic to an op en subset in Rn while preserving the co dimension of faces. Thus, + cone(S ) acquires the structure of a manifold with corners (see [2, Definition 6.13]). At the same time, every p oint of ZS = -1 (cone(S )) lies in one of the subsets {-1 (Uv )}, which is an op en subset in R2n â T m-n . Since the latter is a manifold, so is ZS . A T k -action on the space X is called almost free if all the isotropy subgroups are finite. The toral rank of the space X is the greatest numb er k for which there exists an almost free T k -action on X . Denote the toral rank by trk(X ). The following theorem was proved in [6, § 7.1] in the case of a simplicial complex S ; however, the arguments also apply to the general case. Theorem 6.4. Let S be a simplicial cel l complex of dimension n - 1 with m vertices. Then, trk ZS m - n. Pro of. Let us cho ose a linear system of parameters t1 ,... ,tn in the ring Q[S ] according to Lemma 5.5. Write ti = i1 v1 + ... + im vm , i = 1,... ,n; then, the matrix (ij ) defines a linear mapping : Qm Qn . Replacing, if necessary, by k with a sufficiently large k, we can assume that the mapping is induced by the mapping Zm Zn , which we will also denote by . It follows from Theorem 5.4 that, for any simplex S , the restriction |Zver : Zver Zn of the mapping onto the co ordinate subspace Zver Zm is injective. Denote by N a subgroup in T m that corresp onds to the kernel of the mapping : Zm Zn . Then, N is a pro duct of an (m - n)-dimensional torus and a finite group, and N intersects co ordinate subgroups of the form T ver T m along finite subgroups. Prop osition 6.2 implies that N acts on ZS almost freely; this completes the pro of. The following toral rank conjecture is well-known (Halp erin [7]): the inequality dim H (X ; Q) 2tr
k(X )

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holds for any finite-dimensional space X . This conjecture has b een proved in many particular cases. Our approach provides a rich class of torus actions. Theorem 6.4 implies that if the toral rank conjecture holds, then dim H (ZS ; Q) 2m-n . In the case of a simplicial complex K , we proved in [1] that H (ZK ; Q) =
[m]

H (K ; Q),

where K is a full sub complex in K spanned over the vertex subset [m]. (A stronger result was obtained [2, Corollary 8.8].) Corollary 6.5. Under the toral rank conjecture, the fol lowing inequality holds for any simplicial complex K : dim
[m]

H (K ; Q) 2m-n .

This assertion, concerning the combinatorial structure of simplicial complexes, has b een confirmed in a numb er of cases. For free torus actions, the arguments used in the pro of of Theorem 6.4 lead to the following prop osition. Theorem 6.6. In the torus T m , a toral subgroup of dimension m - n that acts freely on ZS exists if and only if the ring Z[S ] admits a linear system of parameters. The maximal numb er k for which there exists a toral subgroup of dimension k in T m that acts freely on ZS is denoted by s(S ) and is an imp ortant combinatorial invariant of a simplicial cell complex S (for the case of simplicial complexes, see the discussion in [2, § 7.1]). Note that, for simplicial complexes, the diagonal circle subgroup in T m acts on ZK freely (indeed, we have m > n, and therefore the diagonal subgroup intersects each isotropy subgroup T ver only at unity). Thus, s(K ) 1, and the Euler characteristic of the space ZK is equal to zero. Example 6.1 shows that this is not the case for simplicial cell complexes S . ACKNOWLEDGMENTS We are grateful to G. Lupton, who turned our attention to the toral rank conjecture, and to E.V. Shchepin for a helpful suggestion that was used in the pro of of Theorem 4.1. This work was supp orted by the grant of the President of the Russian Federation (pro ject no. NSh-2185.2003.1) and by the Russian Foundation for Basic Research (pro ject nos. 02-01-00659 and 04-01-00702). REFERENCES
1. V. M. Buchstab er and T. E. Panov, "Torus Actions, Combinatorial Top ology, and Homological Algebra," Usp. Mat. Nauk 55 (5), 3­106 (2000) [Russ. Math. Surv. 55, 825­921 (2000)]. 2. V. M. Buchstab er and T. E. Panov, Torus Actions in Topology and Combinatorics (MTsNMO, Moscow, 2004) [in Russian]. 3. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series (Nauka, Moscow, 1981) [in Russian]. 4. J. W. Alexander, "The Combinatorial Theory of Complexes," Ann. Math. 31, 292­320 (1930). 5. W. Bruns and J. Herzog, Cohen­Macaulay Rings (Cambridge Univ. Press, Cambridge, 1998), Cambridge Stud. Adv. Math. 39. 6. M. W. Davis and T. Januszkiewicz, "Convex Polytop es, Coxeter Orbifolds and Torus Actions," Duke Math. J. 62, 417­451 (1991). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Vol. 247 2004


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7. S. Halp erin, "Rational Homotopy and Torus Actions," in Aspects of Topology (Cambridge Univ. Press, Cambridge, 1985), London Math. Soc. Lect. Note Ser. 93, pp. 293­306 . 8. A. Hattori and M. Masuda, "Theory of Multi-Fans," Osaka J. Math. 40, 1­68 (2003). 9. I. Izmestiev and M. Joswig, "Branched Covering, Triangulations, and 3-Manifolds," math.GT/0108202. 10. V. Klee, "A combinatorial Analogue of Poincar´'s Duality Theorem," Can. J. Math. 16, 517­531 (1964). e 11. M. Masuda and T. Panov, "On the Cohomology of Torus Manifolds," math.AT/0306100. 12. S. V. Matveev, Algorithmic Topology and Classification of 3-Manifolds (Springer, New York, 2003). 13. R. P. Stanley, Enumerative Combinatorics (Wadsworth and Brooks/Cole, Monterey, CA, 1986; Mir, Moscow, 1990), Vol. 1. 14. R. P. Stanley, " f -Vectors and h-Vectors of Simplicial Posets," J. Pure Appl. Algebra 71, 319­331 (1991). 15. R. P. Stanley, Combinatorics and Commutative Algebra , 2nd ed. (Birkh¨ ser, Boston, 1996), Progr. Math. 41. au

Translated by I. Nikitin

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