Optimal sweepouts of a Riemannian 2-sphere

Given a sweepout of a Riemannian 2-sphere which is composed of curves of length less than $L$, we construct a second sweepout composed of curves of length less than $L$ which are either constant curves or simple curves. This result, and the methods used to prove it, have several consequences; we answer a question of M. Freedman concerning the existence of min-max embedded geodesics, we partially answer a question due to N. Hingston and H.-B. Rademacher, and we also extend the results of [CL] concerning converting homotopies to isotopies in an effective way.