# Remarks on the geometry and the topology of the loop spaces $H^{s}(S^1, N),$ for $s\leq 1/2.$

Abstract : We first show that, for a fixed locally compact manifold N, the space L 2 (S 1 , N) has not the homotopy type of the classical loop space C ∞ (S 1 , N), by two theorems:-the inclusion C ∞ (S 1 , N) ⊂ L 2 (S 1 , N) is null homotopic if N is connected,-the space L 2 (S 1 , N) is contractible if N is compact and connected. Then, we show that the spaces H s (S 1 , N) carry a natural structure of Frölicher space, equipped with a Riemannian metric, which motivates the definition of Riemannian diffeo-logical space.
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Cited literature [31 references]

https://hal.archives-ouvertes.fr/hal-02285964
Contributor : Jean-Pierre Magnot <>
Submitted on : Friday, September 13, 2019 - 11:49:05 AM
Last modification on : Saturday, September 14, 2019 - 1:40:51 AM

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Jean-Pierre Magnot. Remarks on the geometry and the topology of the loop spaces $H^{s}(S^1, N),$ for $s\leq 1/2.$. International Journal of Maps in Mathematics, 2019, pp.14 - 37. ⟨hal-02285964⟩

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