Some results on semi-free actions of finite groups on Hilbert cube manifolds

Abstract

In this note, we will prove the following results 1. Theorem 1. Given a tame semi-free action of a finite or compact Lie group on a Hilbert cube manifold whose fixed point set is a locally flat submanifold (codimension finite or infinite), if the orbit space is an absolute neighborhood retract, then it is a Hilbert cube manifold. 2. Theorem 2. The homeomorphism group of the orbit space of the standard based semi-free action of a finite group or a torus is locally contractible. 3. Theorem 3. Let a be a fiber preserving semi-free action of a finite group G on Q × [0, 1] n over [0, 1] n with fixed point set {0} × [0, 1] n . If a ¦ Q × t is (weakly) equivalent to the standard semi-free action σ of G on the Hilbert cube Q, then a is fiber preserving (weakly) equivalent to the action σ × Id [0, 1] n . Finally, a generalization of Wong's theorem follows.