Reconstruction of manifolds and subsets of normed spaces from subgroups of their homeomorphism groups

This work concerns with topological spaces of the following types: open subsets of normed vector spaces, manifolds over normed vector spaces, the closures of open subsets of normed vector spaces and some other types of topological spaces related to the above. We show that such spaces are determined by various subgroups of their auto-homeomorphism groups. Theorems 1 3 below are typical examples of the results obtained in this work. Theorem 1 For a metric space X let UC(X) denote the group of all auto-homeomorphisms h of X such that h and h−1 are uniformly continuous. Let X be an open subset of a Banach space with the following property: for every ε > 0 there is δ > 0 such that for every u, v ∈ X: if ‖u− v‖ < δ, then there is an arc L ⊆ X connecting u and v such that diameter (L) < ε. Suppose that the same holds for Y . Let φ be a group isomorphism between UC(X) and UC(Y ). Then there is a homeomorphism τ between X and Y such that τ and τ−1 are uniformly continuous and for every g ∈ UC(X), φ(g) = τ ◦g ◦ τ−1. See Corollaries 5.6 and 2.26. Theorem 2 Let H(X) denote the group of auto-homeomorphisms of a topological space X. Let X be a bounded open subset of a Banach space E, and denote by cl (X) the closure of X in E. Suppose that X has the following properties: (1) There is d such that for every u, v ∈ X there is a rectifiable arc L ⊆ X connecting u and v such that length (L) < d; such that length(L) < d; (2) for every point w in the boundary of X and ε > 0, there is δ > 0 such that for every u, v ∈ X: if ‖u−w‖, ‖v−w‖ < δ, then there is an arc L ⊆ X connecting u and v such that diameter (L) < ε. Suppose that the same holds for Y . Let φ be a group isomorphism between H(cl (X)) and H(cl (Y )). Then there is a homeomorphism τ between cl (X) and cl (Y ) such that for every g ∈ H(cl (X)), φ(g) = τ ◦g ◦τ−1. See Theorems 6.22 and 6.3(b) and Proposition 6.2(c). Theorem 3 Let LIP(X) denote the group of bilipschitz auto-homeomorphisms of a metric space X. Suppose that F,K are the closure of bounded open subsets of Rn, and suppose further that F,K are manifolds with boundary with an atlas consisting of bilipschitz charts. Let φ be a group isomorphism between LIP(F ) and LIP(K). Then there is a bilipschitz homeomorphism τ between F and K such that φ(g) = τ ◦g ◦ τ−1 for every g ∈ LIP(F ). See Corollary 8.5.