The topology of moment angle complexes

Abstract

The purpose of this thesis is to study some topological properties of moment angle complexes of simplicial posets. In case of simplicial complex, a lot of topological properties were studied by other mathematicians. Taras Panov is one of the important contributors. Later, Zhi $L\uuml$ and Taras Panov generalized this concept to the case of simplicial posets. Their main result was that the integral cohomology ring of the moment angle complex of simplicial poset can be obtained by the Tor-algebra of face ring of the given simplicial poset. The first part of this thesis is devoted to the study of simplicial posets and some backgrounds about the homological algebra related with the face rings of simplicial posets. In particular Tor-algebra of face ring and Koszul resolution of $\mathbb{Z}$ are the most important tools to calculate the cohomology rings of the moment angle complexes. In the second part, we give the topology of the moment angle complexes $\mathcal{Z}_{\mathcal{S}}$. Assuming that the simplicial poset $\It{S}$ is defined on the vertex set [$\It{m}$], $\mathcal{Z}_{\mathcal{S}}$ is actually a subspace of unit polydisk $(D^{2})^{m}$. So there is a natural cell structure of the moment angle complex induced from that of $(D^{2})^{m}$. So, we can compose the cellular cochain complex of $\mathcal{Z}_{\mathcal{S}}$, and finally calculate the its cohomology, which is closely related with the Tor-algebra of face ring of $\S$. Using the result of Zhi L$ddot{u} and Taras Panov, we prove some properties of multigraded Betti numbers of $Z_{S}$, and Poincar$acute{e} duality in the multigraded sense. We also investigate Euler characteristic of $\mathcal{Z}_{\mathcal{S}}$ by using some combinatorial arguments. Finally, we study the orbit spaces of moment angle manifolds and their face rings in the last part. The orbit spaces of the moment angle manifolds of simplicial complexes are known to be combinatorially equivalent to simple polytopes. However, in case of s...