On Murayama's theorem on extensor properties of G-spaces of given orbit types (Q409673)

A classical result of J. Dugundji states that for a metrizable space the ANR-property is characterized by the property of admitting arbitrarily fine dominations by simplicial complexes. In equivariant topology the role of simplicial complexes is played by \(G\)-CW-complexes. The following conjecture is stated.NEWLINENEWLINE\textbf{Conjecture 1.1} Let \(G\) be a compact group. Then any metrizable \(G\)-space belonging to \(G\)-ANE admits arbitrarily fine dominations by \(G\)-CW-complexes.NEWLINENEWLINEThe authors point out that the conjecture has been resolved affirmatively for compact metrizable \(G\)-ANE-spaces and for metrizable \(G\)-ANE-spaces with an action by a compact \(0\)-dimensional group. But in general the conjecture has not been verified.NEWLINENEWLINEIn this paper a method of extending actions of compact transformation groups is developed. It is then applied to the problem of preservation of the equivariant extension property by passing to a subspace of given orbit types.