On the formality of equivariant classifying spaces (Q1346352)

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.