Approximating equivariant mapping spaces (Q1175100)

Let \(G\) be a finite group and let \(V\) be a finite-dimensional unitary representation of \(G\) with unit sphere \(SV\). Let \({\mathcal U}(SV,SV)\) be the \(G\)-space of unbased maps \(SV\to SV\), where \(G\) acts by conjugation. Let also \(S^ V\) denote the one-point compactification of \(V\) and let \(\Omega^ VS^ V\) denote the \(G\)-space of based maps \(S^ V\to S^ V\). The authors describe \(G\)-homotopy approximations for the loop-spaces \({\mathcal U}(SV,SV)\) and (when \(V\) is a real orthogonal representation with a trivial summand) \(\Omega^ VS^ V\). As applications they obtain splittings of unstable equivariant mapping spaces in the based and unbased case, along the lines of tom Dieck. Secondly they show that \(\hbox{colim}_ V{\mathcal U}(SV,SV)\) possesses an equivariant Hopf space structure compatible with addition of homotopy classes, the colimit being taken over a suitable collection of unitary representations \(V\) with \(V^ G=0\). For their main results the authors invoke the machinery of \(G\)-quasifibrations, using the Dold-Thom criterion for quasifibration in a form given by May.