Geodesic
Decomposable Free Loop Spaces
Canadian journal of mathematics, Tome 39 (1987) no. 4, pp. 938-955

Voir la notice de l'article provenant de la source Cambridge University Press

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 S 1 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 S 1 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. 
Aguadé, J. Decomposable Free Loop Spaces. Canadian journal of mathematics, Tome 39 (1987) no. 4, pp. 938-955. doi: 10.4153/CJM-1987-047-9
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