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 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 compact-open 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 .

**[An]**R. D. Anderson,*Spaces of homeomorphisms of finite graphs*, Illinois J. Math. (to appear).**[An-Sc]**R. D. Anderson and R. Schori,*Factors of infinite-dimensional manifolds*, Trans. Amer. Math. Soc.**142**(1969), 315–330. MR**0246327**, 10.1090/S0002-9947-1969-0246327-5**[Ce]**Jean Cerf,*Groupes d’automorphismes et groupes de difféomorphismes des variétés compactes de dimension 3*, Bull. Soc. Math. France**87**(1959), 319–329 (French). MR**0116351****[Ch]**T. A. Chapman,*Homeomorphisms of Hilbert cube manifolds*, Trans. Amer. Math. Soc.**182**(1973), 227–239. MR**0372863**, 10.1090/S0002-9947-1973-0372863-8**[Ch-Wo, II]**T. A. Chapman and R. Y. T. Wong,*On homeomorphisms of infinite dimensional bundles. II*, Trans. Amer. Math. Soc.**191**(1974), 261–268; ibid. 191 (1974), 269–276. MR**0415626**, 10.1090/S0002-9947-1974-0415626-8**[Do]**T. Dobrowolski, correspondence, 1982.**[Du]**James Dugundji,*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606****[Ed-Ki]**Robert D. Edwards and Robion C. Kirby,*Deformations of spaces of imbeddings*, Ann. Math. (2)**93**(1971), 63–88. MR**0283802****[Fe]**Steve Ferry,*The homeomorphism group of a compact Hilbert cube manifold is an 𝐴𝑁𝑅*, Ann. of Math. (2)**106**(1977), no. 1, 101–119. MR**0461536****[Ge]**Ross Geoghegan,*On spaces of homeomorphisms, embeddings and functions. I*, Topology**11**(1972), 159–177. MR**0295281****[He]**David W. Henderson,*Infinite-dimensional manifolds are open subsets of Hilbert space*, Topology**9**(1970), 25–33. MR**0250342****[Lu-Ma]**R. Luke and W. K. Mason,*The space of homeomorphisms on a compact two-manifold is an absolute neighborhood retract*, Trans. Amer. Math. Soc.**164**(1972), 275–285. MR**0301693**, 10.1090/S0002-9947-1972-0301693-7**[Re]**Peter L. Renz,*The contractibility of the homeomorphism group of some product spaces by Wong’s method*, Math. Scand.**28**(1971), 182–188. MR**0305426****[T]**H. Toruńczyk,*Absolute retracts as factors of normed linear spaces*, Fund. Math.**86**(1974), 53–67. MR**0365471****[T]**-,*Homeomorphism groups of compact Hilbert cube manifolds which are manifolds*, Bulletin, Polish Academy of Science, 1977.**[We]**James E. West,*Cartesian factors of infinite-dimensional spaces*, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Springer, Berlin, 1974, pp. 249–268. Lecture Notes in Math., Vol. 375. MR**0368014****[Wo]**Raymond Y. T. Wong,*On homeomorphisms of certain infinite dimensional spaces*, Trans. Amer. Math. Soc.**128**(1967), 148–154. MR**0214040**, 10.1090/S0002-9947-1967-0214040-4

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1983-0718999-2

Keywords:
Hilbert manifolds,
spaces of homeomorphisms,
absolute neighborhood extensors,
-sets

Article copyright:
© Copyright 1983
American Mathematical Society