Remarks on homotopy equivalence of configuration spaces of a polyhedron
type: Item
instance of: scholarly article
title: Remarks on homotopy equivalence of configuration spaces of a polyhedron
zbMATH Open document ID: 1437.55022
author: Leonid Plachta
published in: European Journal of Mathematics
publication date: 2020-04-15T00:00:00Z
review text: For a given topological space, \(F_k(X)\) denotes \textit{the configuration space} of \(k\) particles in \(X\) without collisions, i.e., \(F_k(X)=\{(x_1\ldots,x_k)\in X^k:~x_i\neq x_j \text{ for } i\neq j\}\). The \textit{symmetric group} \(\Sigma_k\) acts freely on \(F_k(X)\) by permuting coordinates of \(X^k=X\times\cdots\times X\). The main result in the present article, Theorem 2.2, is to prove the boundary homotopy invariance of the configuration spaces for polyhedra. Explicitly, the author proves that the configuration space \(F_k(Q)\) is \(\Sigma_k\)-equivariantly homotopy equivalent to the configuration space \(F_k(Q\setminus P)\), for any pair of polyhedra \((Q,P)\) where \(P\) is a compact subpolyhedron of \(Q\) that has a PL collar in \(Q\). Furthermore, in the last part of the paper, the author present a question of cell approximation, up to homotopy, of the configuration spaces for PL manifolds via \textit{discrete configuration spaces}.
reviewed by: Cesar A. Ipanaque Zapata
Mathematics Subject Classification ID: 55R80
Mathematics Subject Classification ID: 57Q91
zbMATH DE Number: 7189616
zbMATH Keywords: configuration space
zbMATH Keywords: PL manifold
zbMATH Keywords: deformation retraction
zbMATH Keywords: equivariant map
zbMATH Keywords: polyhedron
zbMATH Keywords: collar
zbMATH Keywords: discrete configuration space
MaRDI profile type: Publication
full work available at URL: s40879-018-00309-0
OpenAlex ID: W2903655514
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DOI: 10.1007/S40879-018-00309-0
Wikidata QID: Q128708094
description: scientific article; zbMATH DE number 7189616