On the formality of equivariant classifying spaces
type: Item
instance of: scholarly article
title: On the formality of equivariant classifying spaces
zbMATH Open document ID: 0813.55007
DOI: 10.1007/BF02571905
author: Mel Rothenberg
author: Georgia Visnjic Triantafillou
published in: Mathematische Zeitschrift
publication date: 1995-06-12T00:00:00Z
full work available at URL: https://eudml.org/doc/174730
review text: Let \(G\) be a finite group and let \(\alpha : G \to U(n)\) be a representation of \(G\). In this note we introduce a \(G\)-space \(BU(\alpha)\) and a \(G\)-vector bundle over it which classifies a certain class of \(G\)- vector bundles modeled by the representation \(\alpha\). We study the equivariant homotopy type of the classifying \(G\)-space \(BU (\alpha)\) and in particular we show that it is equivariantly formal for any finite group \(G\), i.e. the rational \(G\)-homotopy type of \(BU (\alpha)\) is determined by the rational cohomology of its fixed point sets. Since the fixed point sets are products of \(BU (m)\)'s their cohomology can be easily computed. The computation of \(BU (\alpha)\) yields information on the question of equivariant characteristic classes as we show in the applications. The general classifying space \(BU_ n (G)\) which classifies complex \(G\)-vector bundles is also shown to be equivariantly formal.
Mathematics Subject Classification ID: 55R91
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 737188
zbMATH Keywords: \(G\)-vector bundles modeled by a representation
zbMATH Keywords: classifying \(G\)-space
zbMATH Keywords: rational \(G\)-homotopy type
zbMATH Keywords: rational cohomology of fixed point sets
zbMATH Keywords: finite group
zbMATH Keywords: representation
zbMATH Keywords: equivariantly formal
zbMATH Keywords: equivariant characteristic classes
MaRDI profile type: Publication
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OpenAlex ID: W2047048636
description: scientific article; zbMATH DE number 737188