Polyhedral products for connected sums of simplicial complexes
A simplicial complex \(K\) on the vertex set \([m]=\{1,2,\dots ,m\}\) is a collection of subset \(\sigma\subset [m]\) such that for any \(\sigma\in K\) all subsets of \(\sigma\) also belong to \(K\). For the sequence of CW-pairs \((\underline{X},\underline{A})=\{(X_i,A_i)\}_{i=1}^m\) and a simplicial complex \(K\) on the vertex set \([m]\), let \((\underline{X},\underline{A})^K\) denote the polyhedral product defined by \((\underline{X},\underline{A})^K =\bigcup_{\sigma\in K}(\underline{X},\underline{A})^{\sigma}\), where \[ (\underline{X},\underline{A})^{\sigma}=\prod_{i=1}^mY_i, \quad Y_i= \begin{cases} X_i & \text{ if }i\in \sigma, \\ A_i & \text{ if }i\notin \sigma. \end{cases} \] In this paper, the author studies the homotopy type of the polyhedral products for connected sums of simplicial complexes. More precisely, he proves the following more general result: Theorem. Let \(1\leq l<m<n\), and let \(K\) and \(L\) be simplicial complexes on the vertex sets \(\{1,2,\dots ,m\}\) and \(\{l+1,l+2,\dots ,n\}\), respectively. Let \(M\) be a subcomplex of both \(K\) and \(L\) on the vertex set \(\{l+1,l+2,\dots ,m\}\). \begin{itemize} \item[(i)] Then there is a homotopy fibration sequence \[ (\mathcal{A}*\mathcal{A}^{\prime})\vee (G\rtimes\mathcal{A}^{\prime})\vee (\mathcal{A}\ltimes H) \stackrel{}{\longrightarrow} (\underline{CA},\underline{A})^{K\cup_{M}L} \stackrel{r}{\longrightarrow} (\underline{CA},\underline{A})^M, \] where \((\underline{CA},\underline{A})=\{(CA_i,A_i)\}_{i=1}^n\), \(\mathcal{A}=\prod_{i=1}^lA_i\), \(\mathcal{A}^{\prime}=\prod_{j=m+1}^nA_j\), and \(G\) and \(H\) are homotopy fibres of the composite of maps \[ \begin{cases} (\underline{CA},\underline{A})^K \stackrel{}{\longrightarrow} (\underline{CA},\underline{A})^{K\cup_{M}L} \stackrel{r}{\longrightarrow} (\underline{CA},\underline{A})^M, \\ (\underline{CA},\underline{A})^L \stackrel{}{\longrightarrow} (\underline{CA},\underline{A})^{K\cup_{M}L} \stackrel{r}{\longrightarrow} (\underline{CA},\underline{A})^M, \end{cases} \] respectively. \item[(ii)] Moreover, this fibration splits after looping and there is a homotopy equivalence \[ \Omega (\underline{CA},\underline{A})^{K\cup_{M}L} \simeq \Omega (\underline{CA},\underline{A})^M \times \Omega ((\mathcal{A}*\mathcal{A}^{\prime})\vee (G\rtimes\mathcal{A}^{\prime})\vee (\mathcal{A}\ltimes H)). \] \end{itemize} In particular, we consider the following case: Let \(K\) and \(L\) be simplicial complexes and let \(\sigma\) be a facet of both \(K\) and \(L\). Let \(K^{\prime}\) (resp. \(L^{\prime}\)) be the subcomplex of \(K\) (resp. subcomplex of \(L\)) consisting of all simplices of \(K\) (resp. \(L\)) except \(\sigma\). Then the connected sum \(K\#_{\sigma}L\) is given by \(K\#_{\sigma}L=K^{\prime}\cup_{\partial \sigma}L^{\prime}.\) Thus, it follows from the above result that we easily obtain the following fibration sequence \[ (\mathcal{A}*\mathcal{A}^{\prime})\vee (G\rtimes\mathcal{A}^{\prime})\vee (\mathcal{A}\ltimes H) \stackrel{}{\longrightarrow} (\underline{CA},\underline{A})^{K\#_{\partial \sigma}L} \stackrel{r}{\longrightarrow} (\underline{CA},\underline{A})^{\partial \sigma}. \] Furthermore, we also obtain the homotopy equivalence \[ \Omega (\underline{CA},\underline{A})^{K\#_{\partial \sigma}L} \simeq \Omega (\underline{CA},\underline{A})^{\partial \sigma} \times \Omega ((\mathcal{A}*\mathcal{A}^{\prime})\vee (G\rtimes\mathcal{A}^{\prime})\vee (\mathcal{A}\ltimes H)). \] The proof is based on the careful analysis of several homotopy pushouts and homotopy pullbacks, and the author also gives several interesting examples of the above results.
scientific article; zbMATH DE number 7618850
scientific article