Converting homotopies to isotopies and dividing homotopies in half in an effective way

Abstract

We prove two theorems about homotopies of curves on 2-dimensional Riemannian manifolds. We show that, for any epsilon > 0, if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by L + epsilon. If the manifold is orientable, then for any epsilon > 0 we show that, if we can contract a curve gamma traversed twice through curves of length bounded by L, then we can also contract gamma through curves bounded in length by L + epsilon. Our method involves cutting curves at their self-intersection points and reconnecting them in a prescribed way. We consider the space of all curves obtained in this way from the original homotopy, and use a novel approach to show that this space contains a path which yields the desired homotopy.