Spaces of maps into classifying spaces for equivariant crossed complexes
type: Item
instance of: scholarly article
title: Spaces of maps into classifying spaces for equivariant crossed complexes
zbMATH Open document ID: 0898.55009
DOI: 10.1016/S0019-3577(97)89117-3
author: Ronald Brown
author: Marek Golasiński
author: Timothy Porter
author: Andrew Tonks
published in: Indagationes Mathematicae. New Series
publication date: 1998-10-25T00:00:00Z
review text: A crossed complex is an algebraic structure which can be used in homotopy theory; roughly speaking, it is a chain complex of groupoids. A filtered topological space \(X_*\) has a fundamental crossed complex \(\pi X_*\), and a crossed complex \(C\) has a classifying space \(BC\). If \(X_*\) is the skeletal filtration of a CW-complex \(X\) and if \(C\) is a crossed complex, then there is a weak equivalence \[ B\bigl( {\mathcal CRS} (\pi X_*,C) \bigr) \to {\mathcal TOP} (X,BC), \] where \({\mathcal CRS}\) is an internal hom for crossed complexes and \({\mathcal TOP}\) is a space of maps. The main result in this paper is a similar equivariant result for a group \(G\) which is discrete, or at least close to being discrete. The CW-complex becomes a \(G\)-CW-complex, and the crossed complex becomes a diagram of crossed complexes indexed by the orbit category of \(G\). Applications include results on equivariant Eilenberg-MacLane spaces, on local systems, and on equivariant \(2\)-types.
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 20L05
Mathematics Subject Classification ID: 55U15
zbMATH DE Number: 1052948
zbMATH Keywords: equivariant crossed complexes
reviewed by: Richard John Steiner
MaRDI profile type: Publication
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full work available at URL: s0019-3577(97)89117-3
OpenAlex ID: W2136764192
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description: scientific article; zbMATH DE number 1052948