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

TL;DR

This paper investigates the topology of loop spaces with Sobolev regularity, revealing differences from classical loop spaces and introducing a Riemannian Fr"olicher space structure for these Sobolev loop spaces.

Contribution

It demonstrates that Sobolev loop spaces differ topologically from classical loop spaces and introduces a Riemannian Fr"olicher space framework for these spaces.

Findings

The inclusion of smooth loops into Sobolev loops is null homotopic for connected N.

Sobolev loop spaces are contractible when N is compact and connected.

Sobolev loop spaces can be endowed with a natural Fr"olicher space and Riemannian metric.

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 Fr\"olicher space, equipped with a Riemannian metric, which motivates the definition of Riemannian Fr\"olicher space.