On "small geodesics" and free loop spaces

TL;DR

This paper constructs a topological group model for the free loop space of a Riemannian manifold using composable small geodesics, providing new geometric and topological insights into loop spaces.

Contribution

It introduces a novel model for free loop spaces based on composable small geodesics, extending Milnor's approach to Riemannian manifolds.

Findings

Constructed a topological group homotopy equivalent to the loop space of M.

Developed models for free loop space and surface maps using small geodesics.

Established an analogy with Milnor's free group construction.

Abstract

A topological group is constructed which is homotopy equivalent to the pointed loop space of a path-connected Riemannian manifold $M$ and which is given in terms of "composable small geodesics" on $M$. This model is analogous to J. Milnor's free group construction \cite{Milnor} which provides a model for the pointed loop space of a connected simplicial complex. Related function spaces are constructed from "composable small geodesics" which provide models for the free loop space of $M$ as well as the space of continuous maps from a surface to $M$.