Remarks on the geometry and the topology of the loop spaces Hˢ(S¹, N), for s≤ 1/2.

Original · EN

We first show that, for a fixed locally compact manifold N, the space L²(S¹,N) has not the homotopy type odf the classical loop space C∞(S¹,N), by two theorems: - the inclusion C∞(S¹,N) ⊂ L²(S¹,N) is null homotopic if N is connected, - the space L²(S¹,N) is contractible if N is compact and connected. After this first remark, we show that the spaces Hˢ(S¹,N) carry a natural structure of Frölicher space, equipped with a Riemannian metric, which motivates the definition of Riemannian Frölicher space.