Polyhedral products for connected sums of simplicial complexes
type: Item
instance of: scholarly article
title: Polyhedral products for connected sums of simplicial complexes
zbMATH Open document ID: 1505.55021
DOI: 10.1134/S0081543822020080
author: Stephen Theriault
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2022-11-17T00:00:00Z
review text: 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},\und…
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 55P15
zbMATH DE Number: 7618850
zbMATH Keywords: polyhedral product
zbMATH Keywords: connected sum
zbMATH Keywords: simplicial complex
zbMATH Keywords: facet
zbMATH Keywords: moment-angle manifold
MaRDI profile type: Publication
full work available at URL: s0081543822020080
OpenAlex ID: W4313133924
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description: scientific article; zbMATH DE number 7618850