On the reconstruction of topological spaces from their groups of homeomorphisms

For various classes K of topological spaces we prove that if X₁, X₂ 5 K and X₁, X₂ have isomorphic homeomorphism groups, then X₁ and X₂ are homeomorphic. Let G denote a subgroup of the group of homeomorphisms H(X) of a topological space X. A class K of (X, G) 's is faithful if for every (X₁, G₁), (X₂, G₂) 5 K, if φ: G₁ → G₂ is a group isomorphism, then there is a homeomorphism h between X₁ and X₂ such that for every g 5 G₁ φ (g) = hgh⁻¹. Theorem 1: The following class is faithful: ((X, H(X))|(X is a locally finite-dimensional polyhedron in the metric or coherent topology or I is a Euclidean manifold with boundary) and for every x 5 X x is an accumulation point of (g(x)|g 5 H(X))) ∪ ((X, G)|X is a differentiable or a PL-manifold and G contains the group of differentiable or piecewise linear homeomorphisms) ∪((X, H(X))|X is a manifold over a normed vector space over an ordered field). This answers a question of Whittaker [W], who asked about the faithfulness of the class of Banach manifolds. Theorem 2: The following class is faithful: ((X, G)|X is a locally compact Hausdorff space and for every open T ⊆ X and x 5 T (g(x) | g 5 H(X) and g|(X - T) = Id) is somewhere dense). Note that this class includes Euclidean manifolds as well as products of compact connected Euclidean manifolds. Theorem 3: The following class is faithful: ((X, H(X))|(1) X is a O-dimensional Hausdorff space; (2) for every x 5 X there is a regular open set whose boundary is (x); (3) for every x 5 X there are g₁, g₂ 5 G such that x ≠ g₁(x) ≠ g₁(x) ≠ x, and (4) for every nonempty open V ⊆ X there is g 5 H(X)- (Id) such that g | (X - V) = Id). Note that (2) is satisfied by O-dimensional first countable spaces, by order topologies of linear orderings, and by normed vector spaces over fields different from R. Theorem 4: We prove (Theorem 2.23.1) that for an appropriate class K^T of trees ((Aut(T), T;<, o, Op) | T 5 K^T) is first-order interpretable in (Aut(T) | T 5 K^T).