4/7/2011

Taras Panov
Moscow State University

Moment-angle complexes from simplicial posets

The construction of moment-angle complexes may be extended from simplicial complexes to simplicial posets. As a result, a certain T^m-space Z_S is associated to an arbitrary simplicial poset S on m vertices. Face rings Z[S]$ of simplicial posets generalise those of simplicial complexes, but have much more complicated algebraic structure. These rings Z[S] may be studied by topological methods. The space Z_S has many important topological properties of the original moment-angle complex Z_K associated to a simplicial complex K. In particular, the integral cohomology algebra of Z_S is isomorphic to the Tor-algebra of the face ring Z[S]. This leads directly to a generalisation of Hochster's theorem, expressing the algebraic Betti numbers of the ring Z[S] in terms of the homology of full subposets in S. Finally, the total amount of homology of Z_S may be estimated from below, which settles Halperin's toral rank conjecture for the moment-angle complexes Z_S.