Optimal sweepouts of a Riemannian 2-sphere

Gregory R. Chambers, Yevgeny Liokumovich

Published 2014-11-24Version 1

We prove the following conjecture of R. Rotman. Suppose we are given an epsilon > 0 and a sweepout of a Riemannian 2-sphere which is composed of curves of length at most L. We can then find a second sweepout which is composed of curves of length at most L + epsilon, which are pairwise disjoint, and which are either constant curves or simple curves. We use the techniques involved in proving this statement to partly answer a question due to N. Hingston and H.-B. Rachemacher, and we also use these methods to extend the results of [CL] concerning converting homotopies to isotopies in an effective way.