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Decomposable Free Loop Spaces

Published online by Cambridge University Press:  20 November 2018

J. Aguadé
J. Aguadé*
Affiliation:
Universitat Autònoma de Barcelona, Barcelona, Spain
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In this paper we study the spaces X having the property that the space of free loops on X is equivalent in some sense to the product of X by the space of based loops on X. We denote by ΛX the space of all continuous maps from S1 to X, with the compact-open topology. ΩX denotes, as usual, the loop space of X, i.e., the subspace of ΛX formed by the maps from S1 to X which map 1 to the base point of X.

If G is a topological group then every loop on G can be translated to the base point of G and the space of free loops ΛG is homeomorphic to G × ΩG. More generally, any H-space has this property up to homotopy. Our purpose is to study from a homotopy point of view the spaces X for which there is a homotopy equivalence between ΛX and X × ΩX which is compatible with the inclusion ΩX ⊂ ΛX and the evaluation map ΛX → X.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 4 , 01 August 1987 , pp. 938 - 955
Copyright
Copyright © Canadian Mathematical Society 1987

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