The space of non-degenerate closed curves in a Riemannian manifold

TL;DR

This paper explores the topological relationship between the space of non-degenerate closed loops in a Riemannian manifold and the loop space of the frame bundle, establishing a group completion via localization.

Contribution

It demonstrates that the loop space of the frame bundle is the group completion of the semigroup of non-degenerate loops, achieved through localization with respect to a small twist.

Findings

Omega FTM is the group completion of LM

Omega FTM is obtained by localizing LM with respect to a small twist

The topology of LM relates to the loop space of the frame bundle

Abstract

Let LM be the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three. We compare the topology of LM to that of the loop space Omega FTM on the bundle of frames in the tangent bundle of M. We show that Omega FTM is the group completion of LM, and prove that it is obtained by localizing LM with respect to adding a "small twist".