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

  • Jean-Pierre Magnot LAREMA, university of Angers
Keywords: loop space, loop group, homotopy, diffeology, Fr\


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.