On the Structure of Polyhedral Products

TL;DR

This thesis investigates the structure of polyhedral products associated with simplicial complexes, revealing embeddings, group actions, and spectral sequences that deepen understanding of their topological and algebraic properties.

Contribution

It provides new insights into the structure of polyhedral products, including embeddings, natural transformations, and group actions, with applications to homology and spectral sequences.

Findings

Embedded hypercube graphs in polyhedral products

Described natural transformations of the polyhedral product functor

Analyzed the homology module structure under cyclic group actions

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 $E\mathbb{Z}_n\times_{\mathbb{Z}_n} \mathcal{Z}_{\mathcal{K}_n}(D^1,S^0)$.