On the reconstruction of topological spaces from their groups of homeomorphisms

Matatyahu Rubin

Research output: Contribution to journalArticlepeer-review

41 Scopus citations


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 KT of trees ((Aut(T), T;<, o, Op) | T 5 KT) is first-order interpretable in (Aut(T) | T 5 KT).

Original languageEnglish
Pages (from-to)487-538
Number of pages52
JournalTransactions of the American Mathematical Society
Issue number2
StatePublished - 1 Jan 1989

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics


Dive into the research topics of 'On the reconstruction of topological spaces from their groups of homeomorphisms'. Together they form a unique fingerprint.

Cite this