Thin Loop Groups

TL;DR

This paper demonstrates that for finite simplicial complexes, the space of piecewise linear loops forms a topological group homotopy equivalent to continuous loops, unlike higher loops.

Contribution

It establishes the topological group structure of thin piecewise linear loop spaces and compares their homotopy types with continuous loop spaces.

Findings

Thin piecewise linear loop space is a topological group.

Homotopy type of thin piecewise linear loops matches continuous loops.

Higher loops do not share this property.

Abstract

We verify that for a finite simplicial complex $X$ and for piecewise linear loops on $X$, the "thin" loop space is a topological group of the same homotopy type as the space of continuous loops. This turns out not to be the case for the higher loops.