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

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