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Transactions of the American Mathematical Society

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

 

 

Contracting spaces of maps on the countable direct limit of a space


Author: Richard E. Heisey
Journal: Trans. Amer. Math. Soc. 193 (1974), 389-411
MSC: Primary 54C35
DOI: https://doi.org/10.1090/S0002-9947-1974-0367908-6
MathSciNet review: 0367908
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Abstract: We give conditions sufficient to imply the contractibility of the space of maps, with compact-open topology, on the countable direct limit of a space. Applying these conditions we obtain the following: Let F be the conjugate of a separable infinite-dimensional Banach space with bounded weak-$ ^\ast$ topology, or the countable direct limit of the real line. Then there is a contraction of the space of maps on F which simultaneously contracts the subspaces of open maps, embeddings, closed embeddings, and homeomorphisms. Corollaries of our work are that any homeomorphism on F, F as above, is invertibly isotopic to the identity, and the general linear group of the countable direct limit of lines is contractible.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0367908-6
Keywords: Direct limit, function space, topological vector space, Banach space, bounded weak-$ ^\ast$ topology, Hilbert cube
Article copyright: © Copyright 1974 American Mathematical Society