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Induced Homotopy Equivalences on Mapping Spaces and Duality

Published online by Cambridge University Press:  20 November 2018

Philip R. Heath
Philip R. Heath*
Affiliation:
Memorial University of Newfoundland, St. John's, Newfoundland
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In this paper we present three results involving function constructions, together with an example which shows the usefulness of our considerations. The first result, Theorem 1, states roughly that homotopy is self related through function space constructions. Section 1 is devoted to the precise statement and proof of this theorem.

In order to clarify the results of § 2 we draw an analogy. Recall that under certain mild restrictions on the spaces involved, a map p: E → B is a fibration, and a map i: A → X is a cofibration, if and only if the induced maps p*: EZ → BZ and i*: ZX → ZA are fibrations for all spaces Z. The two theorems of § 2 give, in a more general category, analogous results for the notions of limit and colimit. Finally, in § 3 we suggest the importance of this type of result by using it, together with Theorem 1 and the above analogy, to deduce a result from its dual.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 697 - 700
Copyright
Copyright © Canadian Mathematical Society 1970

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