  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.
