On based-free actions of compact Lie groups on the Hilbert cube
This interesting paper concerns an old problem of the equivalence of any based-free action of a compact Lie group \(G\) on the Hilbert cube \(Q\) to the standard based-free action \(\sigma\) of \(G\) on the countable power \(\widetilde Q=(\operatorname{con} G)^\omega\) of the cone of \(G\) (an action of \(G\) on \(Q\) is based-free if it has a unique fixed point \(*\) outside of which \(G\) acts freely). For a based-free action \(\alpha\) of \(G\) on \(Q\) it is shown that (1) the \(G\)-space \(Q_0=Q\smallsetminus\{*\}\) with the free action \(\alpha\) can be embedded into \(\widetilde Q_0=\widetilde Q\smallsetminus\{*\}\) with the standard action \(\sigma\) as a strong deformation \(G\)-retract; (2) the orbit space \(Q_0/\alpha\) is a strong deformation retract of the orbit space \(\widetilde Q_0/\sigma\); in particular, these orbit spaces have the same homotopy type; (3) the action \(\alpha\) is equivalent to the standard based-free action \(\sigma\) if and only if the orbit space \(Q_0/\alpha\) is a \(Q\)-manifold having the proper homotopy type of the orbit space \(\widetilde Q_0/\sigma\); (4) the action \(\alpha\) is equivalent to the standard action \(\sigma\), provided \(\alpha\) is the diagonal countable product of based-free actions of \(G\) on non-degenerate finite-dimensional AR-compacta. The author also stresses an important role of the cone \(\operatorname{con}G\) of \(G\) in the category of \(G\)-spaces: for any free (resp. compact based-free) \(G\)-space \(X\) equivariant maps \(X\to \operatorname{con}G\) separate points and closed subsets (resp. pairs of points) of \(X\), which implies that any such a space \(X\) admits an equivariant embedding into a power \((\operatorname{con}G)^\tau\) of \(G\).
scientific article; zbMATH DE number 1407510