### 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 - IJMM

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 Field Value Title Remarks on the geometry and the topology of the loop spaces $H^{s}(S^1, N),$ for $s\leq 1/2.$ Creator Magnot, Jean-Pierre Description 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 $C^\infty(S^1,N),$ 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. Then, we show that the spaces $H^s(S^1,N)$ carry a natural structure of Fr\"olicher space, equipped with a Riemannian metric, which motivates the definition of Riemannian diffeological space. Publisher Bayram Şahin Date 2019-03-22 Type info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Format application/pdf Identifier http://www.journalmim.com/index.php/ijmm/article/view/29 Source International Journal of Maps in Mathematics - IJMM; Vol 2 No 1 (2019): International Journal of Maps in Mathematics; 14-37 2636-7467 Language eng Relation http://www.journalmim.com/index.php/ijmm/article/view/29/24 Rights Copyright (c) 2019 International Journal of Maps in Mathematics - IJMM