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

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