On the reconstruction of topological spaces from their groups of homeomorphisms
Author:
Matatyahu Rubin
Journal:
Trans. Amer. Math. Soc. 312 (1989), 487538
MSC:
Primary 54H99; Secondary 20F38, 58B99
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 finitedimensional 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 0dimensional 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 0dimensional 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 firstorder interpretable in .
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 A. M. W. Glass, Y. Gurevich, W. C. Holland, and M. JambuGiraudet, Elementary theory of automorphism groups of doubly homogeneous chains, preprint. MR 619862 (82i:03047)
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 [L2]
 , On the existence of noncomparable homogeneous topologies with the same class of homeomorphisms, Tôhoku Math. J. (2) 22 (1970), 499501. MR 0276944 (43:2684)
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 W. Ling, A classification theorem for manifold automorphism groups, preprint.
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 J. van Mill, Homeomorphism groups and homogeneous spaces, preprint. MR 759672 (86h:54016)
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 D. Maharam, On homogeneous measure algebras, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 108111. MR 0006595 (4:12a)
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 M. Rubin, On the automorphism groups of homogeneous and saturated Boolean algebras, Algebra Universalis 9 (1979), 5486. MR 508669 (80d:03032)
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 , On the reconstruction of complete Boolean algebras from their automorphism groups, Arch. Math. Logik und Grundlagen. 20 (1980), 125146. MR 603333 (82c:06026)
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 , Second countable connected manifolds with elementarily equivalent homeomorphism groups are homeomorphic in the constructible universe, in preparation.
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 M. Rubin and Y. Yomdin, On the reconstruction of smooth manifolds, Banach spaces and measure spaces from their automorphism groups, Israel J. of Math. (to appear).
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 L. Rieger, Some remarks on automorphisms of Boolean algebras, Fund. Math. 38 (1951), 209216. MR 0049863 (14:238a)
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 S. Shelah, Why there are many nonisomorphic models for unsuperstable theories (Proc. Internat. Congr. Math., Vancouver, B. C., 1974, vol. 1), Canadian Math. Congress, Montreal, 1975, pp. 259263. MR 0422015 (54:10008)
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 E. H. Spanier, Algebraic topology, McGrawHill, New York, 1966. MR 0210112 (35:1007)
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 F. Takens, Characterization of a differentiable structure by its group of diffeomorphisms, Bol. Soc. Brasil Mat. 10 (1979), 1726. MR 552032 (82e:58027)
 [W]
 J. V. Whittaker, On isomorphic groups and homeomorphic spaces, Ann. of Math. 78 (1963), 7491. MR 0150750 (27:737)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198909888814
PII:
S 00029947(1989)09888814
Article copyright:
© Copyright 1989
American Mathematical Society
