Polyhedral products for wheel graphs and their generalizations (Q6613467)

Let \(K\) be a simplicial complex on the vertex set \([m]\) and \((X_{i},A_{i})\) be a pair of pointed CW-complexes, where \(A_{i}\subset X_{i}\) is a pointed subspace. The polyhedral product, denoted \((\underline{X}, \underline{A})^{K}\), is defined as:\N\begin{align*}\N(\underline{X}, \underline{A})^{K}:=\bigcup_{\sigma\in K}(\underline{X}, \underline{A})^{\sigma}\subset \prod_{i=1}^{m}X_{i},\N\end{align*}\Nwhere \((\underline{X}, \underline{A})^{\sigma}=\prod_{i=1}^{m}Y_{i}\) where \(Y_{i}=X_{i}\) if \(i\in \sigma\) and \(Y_{i}=A_{i}\) otherwise. If \(X_{i}=D^{2}\) and \(A_{i}=S^{1}\), then the polyhedral product \((\underline{X}, \underline{A})^{K}=\mathcal{Z}_{K}\) is called a moment-angle manifold which is a central object in toric topolgy. One of the key problems in the study of polyhedral products is to determine their homotopy type.\N\NLet \(W_{m}(M)\) be the pushout of the vetetex inclusion into the \(m\)-gon \(V_{m}\to P_{m}\) and the inclusion of \(V_{m}\) into the join of the simplicial cpmplex \(M\), \(V_{m}\to V_{m}*M\). In the paper under review, for \(m\ge 4\), the author determines the homotopy type of the loop space \(\Omega \mathcal{Z}_{W_{m}(M)}\) of the moment-angle complex of \(W_{m}(M)\). In Lemma 5.1 (and more precisely in Theorem 5.9), the author proves that \(\Omega \mathcal{Z}_{W_{m}(M)}\) is homotopy equivalent to\N\begin{align*}\N\Omega \mathcal{Z}_{P_{m}}\times \Omega \mathcal{Z}_{M}\times \Omega (G*H),\N\end{align*}\Nwhere \(G\) is the homotopy fiber of \(\mathcal{Z}_{V_{m}}\to \mathcal{Z}_{P_{m}}\), and \(H=(\prod_{i=m+1}^{n})S^{1}\times \Omega\mathcal{Z}_{M}\). The diffeomorphism type of \(\mathcal{Z}_{P_{m}}\) is known from MacGavran's result, and the homotopy type of \(\mathcal{Z}_{V_{m}}\) is also known from the previous works (including the author's) to be the wedge of spheres. As a corollary, the author shows that \(\Omega \mathcal{Z}_{W_{m}}\) of the wheel graph \(W_{m}\) is homotopy equivalent to a product of spheres and loops on spheres, where the wheel graph \(W_{m}\) is \(W_{m}(M)\) with \(M=\{v\}\).\N\NFor the entire collection see [Zbl 1540.57001].