Orbit projections and \(G-ANR\)-resolutions (Q2901998)

A well known fact from the theory of transformation groups is that if \(G\) is a compact Lie group and \(E\) is a paracompact \(G\)-space with orbits only of one orbit type, then the orbit projection \(p_E:E\to E/G\) is a \(G\)-fibration. In this paper, the authors study the same question but in the context of compact metrizable groups. In that case, they show that the orbit projection \(p_E\) can be approximated by \(G\)-fibrations of \(G-ANR\)'s in a good enough way in the following sense. The orbit projection \(p_E\) admits a \(G-ANR\)-resolution consisting of \(G\)-fibrations \(p_i\) and, moreover, this resolution can be chosen so that each \(p_i\) is an orbit projection \(E_i\to E_i/G\), where each \(E_i\) is a \(G-ANR\).