Moment-angle complexes and combinatorics of simplicial manifolds

Abstract

Let � : (D 2 ) m ! I m be the orbit map for the diagonal action of the torus T m on the unit poly-disk (D 2 ) mC m , I m = (0,1) m is the unit cube. Let C be a cubical subcomplex in I m . The moment- angle complex ma(C) is a T m -invariant bigraded cellular decomposition of the subset � −1 (C) � (D 2 ) m with cells corresponding to the faces of C. Different combinatorial problems concerning cubical complexes and related combinatorial objects can be treated by studying the equivariant topology of corresponding moment-angle complexes. Here we consider moment-angle complexes defined by canonical cubical subdivisions of simplicial complexes. We describe relations between the combinatorics of simplicial complexes and the bigraded cohomology of correspond- ing moment-angle complexes. In the case when the simplicial complex is a simplicial manifold the corresponding moment-angle complex has an orbit consisting of singular points. The complement of an invari- ant neighbourhood of this orbit is a manifold with boundary. The rel- ative Poincare duality for this manifold implies the generalized Dehn- Sommerville equations for the number of faces of simplicial manifolds.