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On homeomorphism spaces of Hilbert manifolds

Author: Raymond Y. Wong
Journal: Proc. Amer. Math. Soc. 89 (1983), 693-704
MSC: Primary 57N20; Secondary 54C55, 58D05
MathSciNet review: 718999
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Abstract: Let $ M$ be a Hilbert manifold modeled on the separable Hilbert space $ {l^2}$. We prove the following Fibred Homeomorphism Extension Theorem: Let $ M \times {B^n} \to {B^n}$ be the product bundle over the $ n$-ball $ {B^n}$, then any (fibred) homeomorphism on $ M \times {S^{n - 1}}$ extends to a homeomorphism on all $ M \times {B^n}$ if and only if it extends to a mapping on all $ M \times {B^n}$. Moreover, the size of the extension may be restricted by any open cover on $ M \times {B^n}$. The result is then applied to study the space of homeomorphisms $ \mathcal{H}(M)$ under various topologies given on $ \mathcal{H}(M)$. For instance, if $ M = {l_2}$ and $ \mathcal{H}({l_2})$ has the compact-open topology, the $ \mathcal{H}({l_2})$ is an absolute extensor for all metric spaces. A counterexample is provided to show that the statement above may not be generalized to arbitrary manifold $ M$.

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  • [An] R. D. Anderson, Spaces of homeomorphisms of finite graphs, Illinois J. Math. (to appear).
  • [An-Sc] R. D. Anderson and R. M. Schori, Factors of infinite-dimensional manifolds, Trans. Amer. Math. Soc. 142 (1969), 315-330. MR 0246327 (39:7631)
  • [Ce] J. Cerf, Groupes d'áutomorphismes et groupes de difféomorphismes des variétés compactes de dimension 3 , Bull. Soc. Math. France 87 (1959), 319-329. MR 0116351 (22:7139)
  • [Ch] T. A. Chapman, Homeomorphisms of Hilbert-cube manifolds, Trans. Amer. Math. Soc. 152 (1973), 227-239. MR 0372863 (51:9067)
  • [Ch-Wo, II] T. A. Chapman and R. Y. Wong, On homeomorphisms of infinite dimensional bundles. II, Trans. Amer. Math. Soc. 191 (1974), 261-268. MR 0415626 (54:3708b)
  • [Do] T. Dobrowolski, correspondence, 1982.
  • [Du] J. Dugundji, Topology, Allyn & Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
  • [Ed-Ki] , R. D. Edwards and R. Kirby, Deformations of space of imbeddings, Ann. of Math. (2) 93 (1971), 63-88. MR 0283802 (44:1032)
  • [Fe] S. Ferry, The homeomorphism group of a compact Hilbert cube manifold is an ANR, Ann. of Math. (2) 106 (1977), 101-119. MR 0461536 (57:1521)
  • [Ge] R. Geoghegan, On spaces of homeomorphisms, embeddings and functions. I, Topology 11 (1972), 159-177. MR 0295281 (45:4349)
  • [He] D. W. Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space, Topology 9 (1970), 25-33. MR 0250342 (40:3581)
  • [Lu-Ma] R. Luke and W. K. Mason, The space of a compact two-manifold is an ANR, Trans. Amer. Math. Soc. 146 (1972), 275-285. MR 0301693 (46:849)
  • [Re] P. Renz', The contractibility of the homeomorphism group of some product spaces by Wong's method, Math. Scand. 28 (1971), 182-188. MR 0305426 (46:4556)
  • [T$ _{01}$] H. Toruńczyk, Absolute retracts as factors of normed linear spaces, Fund. Math. 86 (1974), 53-67. MR 0365471 (51:1723)
  • [T$ _{02}$] -, Homeomorphism groups of compact Hilbert cube manifolds which are manifolds, Bulletin, Polish Academy of Science, 1977.
  • [We] J. E. West, Cartesian factors of infinite-dimensional spaces, Topology Conf. (VPI), Lecture Notes in Math., vol. 375, Springer, Berlin and New York, 1974. MR 0368014 (51:4256)
  • [Wo] R. Y. Wong, On homeomorphisms of certain infinite dimensional spaces, Trans. Amer. Math. Soc. 128 (1967), 148-154. MR 0214040 (35:4892)

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Keywords: Hilbert manifolds, spaces of homeomorphisms, absolute neighborhood extensors, $ {\mathbf{Z}}$-sets
Article copyright: © Copyright 1983 American Mathematical Society

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