On homeomorphism spaces of Hilbert manifolds
Author:
Raymond Y. Wong
Journal:
Proc. Amer. Math. Soc. 89 (1983), 693704
MSC:
Primary 57N20; Secondary 54C55, 58D05
MathSciNet review:
718999
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Abstract: Let be a Hilbert manifold modeled on the separable Hilbert space . We prove the following Fibred Homeomorphism Extension Theorem: Let be the product bundle over the ball , then any (fibred) homeomorphism on extends to a homeomorphism on all if and only if it extends to a mapping on all . Moreover, the size of the extension may be restricted by any open cover on . The result is then applied to study the space of homeomorphisms under various topologies given on . For instance, if and has the compactopen topology, the 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 .
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 T. A. Chapman, Homeomorphisms of Hilbertcube manifolds, Trans. Amer. Math. Soc. 152 (1973), 227239. MR 0372863 (51:9067)
 [ChWo, II]
 T. A. Chapman and R. Y. Wong, On homeomorphisms of infinite dimensional bundles. II, Trans. Amer. Math. Soc. 191 (1974), 261268. MR 0415626 (54:3708b)
 [Do]
 T. Dobrowolski, correspondence, 1982.
 [Du]
 J. Dugundji, Topology, Allyn & Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
 [EdKi]
 , R. D. Edwards and R. Kirby, Deformations of space of imbeddings, Ann. of Math. (2) 93 (1971), 6388. 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), 101119. MR 0461536 (57:1521)
 [Ge]
 R. Geoghegan, On spaces of homeomorphisms, embeddings and functions. I, Topology 11 (1972), 159177. MR 0295281 (45:4349)
 [He]
 D. W. Henderson, Infinitedimensional manifolds are open subsets of Hilbert space, Topology 9 (1970), 2533. MR 0250342 (40:3581)
 [LuMa]
 R. Luke and W. K. Mason, The space of a compact twomanifold is an ANR, Trans. Amer. Math. Soc. 146 (1972), 275285. 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), 182188. MR 0305426 (46:4556)
 [T]
 H. Toruńczyk, Absolute retracts as factors of normed linear spaces, Fund. Math. 86 (1974), 5367. MR 0365471 (51:1723)
 [T]
 , Homeomorphism groups of compact Hilbert cube manifolds which are manifolds, Bulletin, Polish Academy of Science, 1977.
 [We]
 J. E. West, Cartesian factors of infinitedimensional 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), 148154. MR 0214040 (35:4892)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198307189992
PII:
S 00029939(1983)07189992
Keywords:
Hilbert manifolds,
spaces of homeomorphisms,
absolute neighborhood extensors,
sets
Article copyright:
© Copyright 1983
American Mathematical Society
