Remarks on the geometry and the topology of the loop spaces $H^{s}(S^1, N),$ for $s\leq 1/2.$
Keywords:
loop space, loop group, homotopy, diffeology, Fr\Abstract
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.
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Published
2019-03-22
How to Cite
Magnot, J.-P. (2019). 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, 2(1), 14–37. Retrieved from https://simadp.com/journalmim/article/view/29
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