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

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

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 .

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Additional Information

DOI:
https://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