On the Structure of Polyhedral Products

Abstract

In this thesis, we study the structure of the polyhedral product $\mathcal{Z}_{\mathcal{K}}(D^1,S^0)$ determined by an abstract simplicial complex ${\mathcal{K}}$ and the pair $(D^1,S^0)$. We showed that there is natural embedding of the hypercube graph in $\mathcal{Z}_{\mathcal{K}_n}(D^1,S^0)$ where ${\mathcal{K}}_n$ is the boundary of an $n$-gon. This also provides a new proof of a known theorem about genus of the hypercube graph. We give a description of the invertible natural transformations of the polyhedral product functor. Then, we study the action of the cyclic group $\mathbb{Z}_n$ on the space $\mathcal{Z}_{\mathcal{K}_n}(D^1,S^0)$. This action determines a $\mathbb{Z}[\mathbb{Z}_n]$-module structure of the homology group $H_*(\mathcal{Z}_{\mathcal{K}_n}(D^1,S^0))$. We also study the Leray-Serre spectral sequence associated to the homotopy orbit space $E\mathbb{Z}_n\times_{\mathbb{Z}_n} \mathcal{Z}_{\mathcal{K}_n}(D^1,S^0)$.