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

Abstract

We first show that, for a fixed locally compact manifold $N,$ the space $L^2(S^1,N)$ has not the homotopy type odf the classical loop space $C^{\infty }(S^1,N),$ by two theorems: - the inclusion $C^{\infty }(S^1,N)\subset L^2(S^1,N)$ is null homotopic if $N$ is connected, - the space $L^2(S^1,N)$ is contractible if $N$ is compact and connected. After this first remark, 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 Frölicher space.

Enlarged proofs, many typographic errors corrected. arXiv admin note: text overlap with arXiv:1007.3543

References in corpus (1)

• Tangent spaces and tangent bundles for diffeological spaces