Equivariant covering spaces and homotopy covering spaces
type: Item
instance of: scholarly article
title: Equivariant covering spaces and homotopy covering spaces
zbMATH Open document ID: 1075.55005
author: S. R. Costenoble
author: S. Waner
published in: Homology, Homotopy and Applications
publication date: 2005-04-05T00:00:00Z
full work available at URL: https://eudml.org/doc/51797
full work available at URL: volume6-1.htm
review text: The theory of covering spaces is very closely related to the fundamental groupoid in the nonequivariant case for a non-connected base space, and so it is also equivariantly, in the same form as it relates to the fundamental group in the non-equivariant case with connected base space. This paper starts by defining the fundamental grupoid \(\Pi_GX\) following tom Dieck [\textit{T. tom Dieck}, Transformation groups. Studies in Math., vol. 8, (Berlin-New York:) Walter de Gruyter, (1987; Zbl 0611.57002)], as the category whose objets are the equivariant maps \(x:G/H\rightarrow X\) and whose morphisms are the pairs \((\omega,\alpha)\) where \(\alpha:G/H\rightarrow G/K\) is a \(G\)-map and \(\omega\) is an equivalence class of paths \(x\rightarrow y\circ\alpha\) in \(X^H\). The paper is constructed on the basis of proving analogous results to those one finds in the non-equivariant situation. Let us denote by \({\mathcal O}_G\) the orbit category of \(G\) whose objects are the homogeneous spaces \(G/H\) and whose morphisms are \(G\)-maps. The category of the homotopy \(G\)-covering spaces over orbits, that is of constructions of the form \(G\times_HF\) and \(G\)-fiber homotopy equivalences is denoted by \({\mathcal D}_G\), and \(h{\mathcal D}_G\) is the homotopy category over \({\mathcal O}_G\) of \({\mathcal D}_G\). There is a projection map \(\pi:{\mathcal D}_G\rightarrow{\mathcal O}_G\) taking \(G\times_HF\) to \(G/H\). There are two main results in this paper, and they are condensed in the following one. \textbf{Theorem.} Given a \(G\)-space \(X\) of the \(G\)-homotopy type of a \(G\)-\(CW\) complex, \textbf{1.} the equivalence classes of \(G\)-covering spaces over \(X\) are in 1-1 correspondence with the isomorphism classes of discrete representations of the fundamental groupoid; \textbf{2.} the (weak) fiber-homotopy equivalence classes of (weak) homotopy \(G\)-covering spaces are in 1-1 correspondence with the isomorphism classes of homotopy discrete representations o…
Mathematics Subject Classification ID: 55R91
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 22A30
Mathematics Subject Classification ID: 55N25
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 57M10
zbMATH DE Number: 2153121
zbMATH Keywords: equivariant covering spaces
zbMATH Keywords: discrete representations
zbMATH Keywords: discrete actions
reviewed by: Juan Antonio Pérez
MaRDI profile type: Publication
DOI: 10.4310/HHA.2004.V6.N1.A23
description: scientific article; zbMATH DE number 2153121