Equivariant embeddings and compactifications of free \(G\)-spaces
A \(G\)-space is a triple \((G,X,\alpha)\) where \(G\) is a compact Lie group, \(X\) is a topological space, and \(\alpha\) is a continuous action from \(G\times X\) to \(X\). A \(G\)-space \(X\) is free if \(gx=x\) for every \(x\in X\) implies \(g=e\), the identity of \(G\). A \(G\)-compactification of a \(G\)-space \(X\) is a pair \((b_G,b_GX)\) where \(b_G\) is a \(G\)-homeomorphic embedding of \(X\) onto a dense subspace of a compact \(G\)-space \(b_GX\). The collection of all subgroups which are conjugate to a subgroup \(H\) of \(G\) is denoted by \((H)\). \(G_x=\{g\in G:gx= x\}\) is the stabilizer of \(x\). A \(G\)-space \(X\) is of orbit type \((H)\) if \((G_x)= (H)\) for all \(x\in X\). The \(G\)-spaces considered in this paper are all assumed to have a single orbit type \((H)\). An equivariant map \(f\) from a \(G\)-space \(X\) to a \(G\)-space \(Y\) is isovariant if \(G_x=G_{f(x)}\) for all \(x\in X\). A \(G\)-space \(X\) is said to be of Euclidean type if there exists an isovariant map from \(X\) into a Euclidean \(G\)-space. It is verified that the following assertions concerning a \(G\)-space \(X\) are equivalent: (1) \(X\) has a \(G\)-compactification of type \((H)\), (2) \(X\) has an isovariant map in to a finite-dimensional, compact, metrizable \(G\)-space of type \((H)\), (3) \(X\) is of Euclidean type, (4) \(X\) has an isovariant map in to a compact \(G\)-space of type \((H)\), (5) \(X\) has a \(G\)-compactification of type \((H)\) and the same weight as \(X\). The author goes on to derive a result which shows that there is a large class of \(G\)-spaces that admit \(G\)-compactifications of a single orbit type. Specifically, a paracompact space \(X\) is finitistic if every open cover has a refinement \(\omega\) for which there exists a natural number \(n\) such that each \(x\in X\) can belong to at most \(n\) sets in \(\omega\). The author proves that every finitistic \(G\)-space \(X\) has a \(G\)-compactification of the same type \((H)\) and the same weight as \(X\). He goes on to investigate universal finite dimensional compact free \(G\)-spaces, that is, compact free \(G\)-spaces which contain \(G\)-homeomorphic copies of all free \(G\)-spaces which satisfy certain conditions.
scientific article; zbMATH DE number 1883654