Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Uniqueness results for homeomorphism groups

Author: Robert R. Kallman
Journal: Trans. Amer. Math. Soc. 295 (1986), 389-396
MSC: Primary 57S05; Secondary 22A05, 54H05, 54H15, 58D05
MathSciNet review: 831205
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a separable metric manifold and let $ \mathcal{H}(X)$ be the homeomorphism group of $ X$. Then $ \mathcal{H}(X)$ has a unique topology in which it is a complete separable metric group. Similar results are demonstrated for a much wider class of spaces, $ X$, and for many subgroups of the homeomorphism group.

References [Enhancements On Off] (What's this?)

  • [1] R. D. Anderson, The algebraic simplicity of certain groups of homeomorphisms, Amer. J. Math. 80 (1958), 955-963. MR 0098145 (20:4607)
  • [2] S. Banach. Théorie des opérations linéaires, PWN, Warsaw, 1932.
  • [3] P. Chernoff and J. Marsden, On continuity and smoothness of group actions, Bull. Amer. Math. Soc. 76 (1970), 1044-1049. MR 0265510 (42:419)
  • [4] R. P. Filipkiewicz, Isomorphisms between diffeomorphism groups, Ergodic Theory and Dynamical Systems 2 (1982), 159-171. MR 693972 (84j:58039)
  • [5] A. Gleason and R. S. Palais, On a class of transformation groups, Amer. J. Math. 79 (1957), 631-648. MR 0089367 (19:663d)
  • [6] R. R. Kallman, A uniqueness result for the infinite symmetric group, Studies in Analysis, Advances in Math. Suppl. Studies, vol. 4, edited by G. C. Rota, Academic Press, New York, 1978, pp. 321-322. MR 546814 (81c:22007)
  • [7] J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955. MR 0070144 (16:1136c)
  • [8] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [9] G. W. Mackey, Borel structures in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 134-165. MR 0089999 (19:752b)
  • [10] R. D. Mauldin (editor), The Scottish book, Birkhäuser, Boston, Mass., 1981. MR 666400 (84m:00015)
  • [11] J. Palis and W. de Meló, Geometric theory of dynamical systems, Springer-Verlag, New York, 1982. MR 669541 (84a:58004)
  • [12] J. V. Whittaker, On isomorphic groups and homeomorphic spaces, Ann. of Math. (2) 78 (1963), 74-91. MR 0150750 (27:737)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57S05, 22A05, 54H05, 54H15, 58D05

Retrieve articles in all journals with MSC: 57S05, 22A05, 54H05, 54H15, 58D05

Additional Information

Keywords: Complete separable metric group, homeomorphism group, diffeomorphism group, manifold, Cantor set, Hilbert cube
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society