# Moment-angle complexes from simplicial posets

# 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 $Z_S$.