On the boundedness of equivariant homeomorphism groups

Abstract

Given a principal \(G\)-bundle \(\pi:M\to B\), let \(\mathcal{H}_G(M)\) be the identity component of the group of \(G\)-equivariant homeomorphisms on \(M\). The problem of the uniform perfectness and boundedness of \(\mathcal{H}_G(M)\) is studied. It occurs that these properties depend on the structure of \(\mathcal{H}(B)\), the identity component of the group of homeomorphisms of \(B\), and of \(B\) itself. Most of the obtained results still hold in the \(C^r\) category.

Keywords

• principal \(G\)-manifold
• equivariant homeomorphism
• uniformly perfect
• bounded
• \(C^r\) equivariant diffeomorphism