Isomorphisms between groups of equivariant homeomorphisms of $G$-manifolds with one orbit type

TL;DR

This paper proves a reconstruction theorem for free G-manifolds and extends it to G-manifolds with one orbit type, highlighting the specificity and complexity of reconstruction in the category of G-spaces.

Contribution

The paper introduces the first reconstruction results for G-manifolds with one orbit type, generalizing previous results for free G-manifolds and locally trivial bundles.

Findings

Reconstruction theorem for free G-manifolds.

Extension of reconstruction to G-manifolds with one orbit type.

Reconstruction in G-spaces is highly specific and involved.

Abstract

Given a compact Lie group $G$, a reconstruction theorem for free $G$-manifolds is proved. As a by-product reconstruction results for locally trivial bundles are presented. Next, the main theorem is generalized to $G$-manifolds with one orbit type. These are the first reconstruction results in the category of $G$-spaces, showing also that the reconstruction in this category is very specific and involved.