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MOMENT-ANGLE COMPLEXES AND COMBINATORICS OF SIMPLICIAL MANIFOLDS
VICTOR M. BUCHSTABER AND TARAS E. PANOV

Let : (D2 )m I m be the orbit map for the diagonal action of torus T m on the unit poly-disk (D2 )m Cm . Each face of the cube I m = [0, 1]m (viewed as a cubical complex) has the form F
I J

= {(y1 , . . . , ym ) I

m

: yi = 0 if i I , yj = 1 if j J }, /
I J

where I J are two subsets of the index set [m] = {1, . . . , m}. For each face F BI J := -1 (FI J ). If #I = i, #J = j , then BI J (D2 )j -i в T m-j . =

put

Definition 1. Let C be a cubical subcomplex of I m . The moment-angle complex ma(C ) is the T m -invariant decomposition of the subset -1 (|C |) (D2 )m into blocks BI J corresponding to the faces FI J of complex C . Many combinatorial problems concerning cubical complexes may be treated by studying the equivariant topology of moment-angle complexes. In the present paper we realize this approach in the case of cubical complexes determined by simplicial complexes. Let K n-1 be an (n - 1)-dimensional simplicial complex with m vertices, and |K | the corresponding polyhedron. If I = {i1 , . . . , ik } [m] is a simplex of K , then we would write I K . Define the following two cubical subcomplexes of I m : cub(K ) = {F
I J

: J K, I = },

cc(K ) = {F

I J

: J K }.

Lemma 2. As a topological space, the complex cub(K ) is homeomorphic to |K |, while cc(K ) is homeomorphic to the cone | cone(K )|. The cubical complex cc(K ) was introduced in [1] and then studied in [2]. The cubical complex cub(K ) appeared in [3].

'
D

$ r

'
D

$ r

a) T

1

b) T

0

rI

1

&

%

&

%

Denote the moment-angle complexes corresponding to cub(K ) and cc(K ) by WK and ZK respectively. Consider the cellular decomposition of the poly-disk (D2 )m that is obtained by subdividing each factor D2 into 0-dimensional cell 1, 1-dimensional cell T , and 2-dimensional cell D, see Fig. a). Each cell of (D2 )m is a product of cells Di , Ti , 1i , i = 1, . . . , m, i.e., can be written as DI TJ 1[m]\I J , where I , J are disjoint subsets of [m]. Set DI TJ := DI TJ 1[m]\I J . Now it can be easily seen that ZK is cellular subcomplex of (D2 )m consisting of all cells DI TJ such that I K . Lemma 3. The embedding T map to a point.
m

=

-1

(1, . . . , 1) Z

K

is a cel lular map homotopic to the

As it was shown in [2], for any field k there is the following isomorphism of algebras: (1) H (ZK ) Tork[v1 ,...,vm ] k(K ), k = H k(K ) [u1 , . . . , um ], d , =
Partially supp orted by the Russian Foundation for Fundamental Research grant no. 99-01-00090.
1

2

VICTOR M. BUCHSTABER AND TARAS E. PANOV

where k(K ) is the StanleyRaisner ring of complex K , and the differential d is defined by d(vi ) = 0, d(ui ) = vi , i = 1, . . . , m. The Tor-algebra from (1) is naturally a bigraded algebra with bideg(vi ) = (0, 2), bideg(ui ) = (-1, 2). The calculation of the ring H (ZK ) allowed to describe the multiplicative structure in the cohomology of the complement of a coordinate subspace arrangement in Cm [2]. In [1] there was introduced the subcomplex C (K ) k(K ) [u1 , . . . , um ] spanned by monomials uJ and vI uJ such that I J = , I K . It was also shown there that the cohomology of C (K ) is also isomorphic to that of ZK . Denote by C (ZK ) and C (ZK ) the chain and the cochain complex for type a) cellular decomposition of ZK respectively. Theorem 4. Let (DI TJ ) C (ZK ) denote the cel lular cochain dual to the cel l DI TJ ZK . The correspondence vI uJ (DI TJ ) defines a canonical isomorphism of complexes C (K ) and C (ZK ), each of which calculates H (ZK ). The pair (ZK , T m ) acquires a bigraded cellular structure by setting bideg(Di ) = (0, 2), bideg(Ti ) = (-1, 2), bideg(1i ) = (0, 0). Put b-q,2p (ZK , T m ) = dim H-q,2p [C (ZK , T m )]. Consider the new cellular decomposition of the poly-disc (D2 )m that is obtained by subdividing each factor D2 into 5 cells D, T , I , 1, 0, see Fig. b). This allows to introduce a bigraded cellular structure on WK and define the numbers bq,2p (WK ) = dim Hq,2p [C (WK )]. Put (ZK , T m ; t) =
p,q

(-1)q b

-q ,2p

(ZK , T m )t2p ,

( WK ; t ) =
p,q

(-1)q b

q ,2p

(WK )t2p .

Let fi be the number of i-simplices of K , and (h0 , . . . , hn ) the h-vector of K determined from the equation h0 tn + . . . + hn-1 t + hn = (t - 1)n + f0 (t - 1)n-1 + . . . + fn-1 . Theorem 5. Put h(t) = h0 + h1 t + · · · + hn tn . Then (ZK , T m ; t) = (1 - t2 )m-n h(t2 ) - (1 - t2 )m , (WK ; t) = (1 - t2 )m-n h(t2 ) + (-1)n-1 hn (1 - t2 )m . Lemma 6. If |K | S =
n-1

(i.e., K is a simplicial sphere), then Z

K

is a closed manifold.

Suppose now that K is a simplicial manifold. Then the complex ZK generally fails to be a manifold. However, removing from ZK a small neighbourhood U (T m ) of the orbit -1 (1, . . . , 1) T m we obtain manifold WK = ZK \ U (T m ) with boundary = W K = |K | в T m . Theorem 7. The manifold (with boundary) WK is equivariantly homotopy equivalent to the complex WK . There is a canonical homeomorphism of pairs (WK , WK ) (ZK , T m ). The relative Poincarґ duality isomorphisms for W e (WK ; t) = (-1) Corollary 8. Let K h
n-i n-1 i m-n 2m K

n-1

imply

t

(ZK , T m ; 1 ). t

Taking into account Theorem 5 we obtain be a simplicial manifold. Then
n i

- hi = (-1) (hn - 1)

= (-1)i (K

n-1

) - (S

n-1

)

n i

,

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

where (·) denotes the Euler number. Rewriting the equation (8) in terms of the f -vector we come to more complicated equations, which were deduced in [4], [5]. For |K | = S n-1 Corollary 8 gives the classical DehnSommerville equations. In the particular case of PL-manifolds the topological invariance of numbers hn-i - hi (which follows directly from Corollary 8) was firstly observed by Pachner in [6, (7.11)]. The extended version of this article is http://xxx.lanl.gov/abs/math.AT/0005199. The authors wish to express special thanks to Oleg Musin for stimulating discussions and helpful comments, in particular, for drawing our attention to the results of [4] and [6].

MOMENT-ANGLE COMPLEXES AND SIMPLICIAL MANIFOLDS

3

References
[1] V. M. Buchstab er and T. E. Panov. Pro ceedings of the Steklov Institute of Mathematics 225 (1999), 87120. [2] V. M. Buchstab er and T. E. Panov. (Russian), Zap. Nauchn. Semin. POMI 266 (2000), 2950. [3] M. A. Shtan'ko and M. I. Shtogrin. Russian Math. Surveys 47 (1992), no. 1, 267268. [4] B. Chen, M. Yan. Pro ceedings of the Steklov Institute of Mathematics 221 (1998), 305319. [5] V. Klee. Canadian J. Math. 16 (1964), 517531. [6] U. Pachner. Europ ean J. Combinatorics 12 (1991), 129145. Department of Mathematics and Mechanics, Moscow State University, 119899 Moscow, RUSSIA E-mail address : buchstab@mech.math.msu.su tpanov@mech.math.msu.su