Splitting a contraction of a simple curve traversed $m$ times

TL;DR

This paper proves that if a curve traversed multiple times can be contracted within a length bound, then the simple curve itself can also be contracted within the same bound, and discusses open questions on curve homotopies.

Contribution

It establishes a length-preserving property for contractions of multiple traversals of a simple curve on Riemannian surfaces, and explores related open problems.

Findings

Contractibility of m-fold traversed curves implies contractibility of the simple curve within the same length bound.

Provides conditions under which length bounds are preserved in curve homotopies.

States open questions about controlling length and self-intersections in homotopies.

Abstract

Suppose that $M$ is a $2$-dimensional oriented Riemannian manifold, and let $\gamma$ be a simple closed curve on $M$. Let $m \gamma$ denote the curve formed by tracing $\gamma$ $m$ times. We prove that if $m \gamma$ is contractible through curves of length less than $L$, then $\gamma$ is contractible through curves of length less than $L$. In the last section we state several open questions about controlling length and the number of self-intersections in homotopies of curves on Riemannian surfaces.