Equivariant self-homotopy equivalences of product spaces
type: Item
instance of: scholarly article
title: Equivariant self-homotopy equivalences of product spaces
author: Gopal Dutta
author: Debasis Sen
author: Ajay Singh Thakur
published in: Topology and its Applications
publication date: 2023-01-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/2206.00963
review text: Let \(G\) be a finite group. Recall that a based \(G\)-space is \(G\)-space whose base-point is \(G\)-fixed. For pointed \(G\)-spaces \(X\) and \(Y\), let \(\mathrm{map}^G(X,Y)\) denote the space of all based \(G\)-maps from \(X\) to \(Y\), and we denote by \([X,Y]_G\) the space of all based \(G\)-homotopy classes of based \(G\)-maps from \(X\) to \(Y\). Let \(\mathrm{aut}^G(X)\) be the space of all based \(G\)-equivariant self-homotopy equivalences of \(X\), and let \(\mathrm{Aut}^G(X)\) denote the set of based \(G\)-equivariant homotopy classes of \(\mathrm{aut}^G(X)\). Note that this is a group whose multiplication is induced from the composition of maps. For a \(G\)-self-homotopy equivalence \(f:X\times Y\to X\times Y\), we denote by \(f_{XX}\) and \(f_{YY}\) the maps given by \[ f_{ZZ}=p_Z\circ f\circ \iota_Z :Z\to Z \quad \text{for }Z=X,Y, \] where \(p_Z:X\times Y \to Z\) and \(\iota_Z:Z\to X\times Y\) denote the projection and the inclusion for \(Z=X\) or \(Y\). Then a based \(G\)-self-homotopy equivalence \(f\in \mathrm{Aut}^G(X\times Y)\) is called \textit{reducible} if \(f_{XX}\in \mathrm{aut}^G(X\times Y)\) and \(f_{YY}\in \mathrm{aut}^G(X\times Y)\). For \(Z=X\) or \(Y\), let \(\mathrm{Aut}^G_{Z}(X\times Y)\) denote the subgroup of \(\mathrm{Aut}^G(X\times Y)\) given by \[ \mathrm{Aut}^G_{Z}(X\times Y)=\{f\in \mathrm{Aut}^G(X\times Y): p_Z\circ f=p_Z\}. \] Let \(\mathrm{aut}(X)\) be the space of all based self-homotopy equivalences of \(X\) and let \(\mathrm{aut}_1(X)\) denote the path-component of the identity map \(\mathrm{id}_X:X\to X\) of the space \(\mathrm{aut}(X)\). In this paper, the authors study the group \(\mathrm{Aut}^G(X\times Y)\), and they prove that \[ \mathrm{Aut}^G(X\times Y)=\mathrm{Aut}^G_X(X\times Y)\cdot \mathrm{Aut}^G_Y(X\times Y) \] if all \(g\)-self-homotopy equivalences of \(X\times Y\) are \(G\)-reducible. Moreover, under the same assumption as above, they also show that there is a split short exact sequence \[ 1\to [X,\mathrm{…
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55Q05
zbMATH DE Number: 7646061
zbMATH Keywords: equivariant self homotopy equivalences
zbMATH Keywords: \(G\)-reducibility
zbMATH Keywords: product spaces
MaRDI profile type: Publication
OpenAlex ID: W4312081592
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DOI: 10.1016/J.TOPOL.2022.108390
description: scientific article; zbMATH DE number 7646061