Thin loop groups (Q2210178)

A loop \(\gamma\) is a thin loop if there exists a homotopy (rel. the base point) of \(\gamma\) to the trivial loop with the image of the homotopy lying entirely within the image of \(\gamma\). Two loops \(f\) and \(g\) are said to be thin homotopic if there is a sequence of loops \(\gamma_1=f,\gamma_2,\dots, \gamma_n=g\) such that for each \(1\leq i<n\), \(\gamma_i\circ \gamma_{i+1}^{-1}\) is thin. A reparameterization of a path \(\gamma :I=[0,1]\to X\) is any map obtained from \(\gamma\) by precomposition with a continuous map \(\alpha:I\to I\) such that \(\alpha (0)=0,\ \alpha (1)=1\). Note that any two curves that differ by a reparameterization are thin homotopic. Let \(\pi_1^1(X)\) denote the thin loop space defined by \(\pi_1^1(X)=\Omega X/\sim\), where \(\sim\) is thin homotopy. In this paper the authors prove that if \(X\) is a finite connected simplicial complex the space of thin PL loops \(\omega (X)\) is a topological group homotopy equivalent to the base loop space \(\Omega X\). They also discuss several other group models for loop spaces.