Homotopy equivalence between diagrams of spaces
type: Item
instance of: scholarly article
title: Homotopy equivalence between diagrams of spaces
zbMATH Open document ID: 0612.55017
DOI: 10.1016/0022-4049(86)90108-8
author: Emmanuel D. Farjoun
author: Alexander Zabrodsky
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: Any discrete group G can be considered as a small category having a single object and having G itself as the set of endomorphisms (in fact automorphisms) of that single object. An action of G on a space X can then be thought of as a functor from G to Top which takes X as its value on the single object of G. \textit{G. E. Bredon} [Equivariant cohomology theories (Lect. Notes Math. 34) (1967; Zbl 0162.272)] proved that if \(f: X\to Y\) is a G-map between G-CW complexes, then it is a G-homology equivalence if and only if for each subgroup H of G the induced map \(f^ H: X^ H\to Y^ H\) on fixed point sets, is a homotopy equivalence. By a careful analysis of Bredon's result, the authors extend it to apply to diagrams of spaces indexed by an arbitrary small category, D. This requires them to generalize the concepts of orbits and fixed point set to the case of a general diagram. These concepts are then applied to give the generalization of Bredon's result together with an application to diagrams of simplicial complexes.
Mathematics Subject Classification ID: 55P99
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55P91
zbMATH DE Number: 3990393
zbMATH Keywords: G-homology equivalence
zbMATH Keywords: diagrams of spaces
zbMATH Keywords: diagrams of simplicial complexes
reviewed by: Timothy Porter
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90108-8
OpenAlex ID: W2064252521
cites work: Equivariant cohomology theories
cites work: Q3808872
cites work: Q3218817
cites work: Q5550390
cites work: Q3977618
description: scientific article; zbMATH DE number 3990393