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

International audienceWe 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