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

Suppose that $M$ is a $2$-dimensional oriented Riemannian manifold, and let $γ$ be a simple closed curve on $M$. Let $m γ$ denote the curve formed by tracing $γ$ $m$ times. We prove that if $m γ$ is contractible through curves of length less than $L$, then $γ$ 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.

No reviews yet.

No comments yet.