On the reconstruction of topological spaces from their groups of homeomorphisms

Author:
Matatyahu Rubin

Journal:
Trans. Amer. Math. Soc. **312** (1989), 487-538

MSC:
Primary 54H99; Secondary 20F38, 58B99

DOI:
https://doi.org/10.1090/S0002-9947-1989-0988881-4

MathSciNet review:
988881

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Abstract: For various classes of topological spaces we prove that if and have isomorphic homeomorphism groups, then and are homeomorphic. Let denote a subgroup of the group of homeomorphisms of a topological space . A class of 's is faithful if for every , if is a group isomorphism, then there is a homeomorphism between and such that for every . *Theorem* 1: The following class is faithful: is a locally finite-dimensional polyhedron in the metric or coherent topology or is a Euclidean manifold with boundary) and for every is an accumulation point of is a differentiable or a -manifold and contains the group of differentiable or piecewise linear homeomorphisms is a manifold over a normed vector space over an ordered field. This answers a question of Whittaker , who asked about the faithfulness of the class of Banach manifolds. *Theorem* 2: The following class is faithful: is a locally compact Hausdorff space and for every open and and 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: (1) is a 0-dimensional Hausdorff space; (2) for every there is a regular open set whose boundary is ; (3) for every there are such that , and (4) for every nonempty open there is such that . Note that (2) is satisfied by 0-dimensional first countable spaces, by order topologies of linear orderings, and by normed vector spaces over fields different from . *Theorem* 4: We prove (Theorem 2.23.1) that for an appropriate class of trees is first-order interpretable in .

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DOI:
https://doi.org/10.1090/S0002-9947-1989-0988881-4

Article copyright:
© Copyright 1989
American Mathematical Society