Abstract
Suppose that [Formula: see text] is a [Formula: see text]-dimensional oriented Riemannian manifold, and let [Formula: see text] be a simple closed curve on [Formula: see text]. Let [Formula: see text] denote the curve formed by tracing [Formula: see text] [Formula: see text] times. We prove that if [Formula: see text] is contractible through curves of length less than [Formula: see text], then [Formula: see text] is contractible through curves of length less than [Formula: see text]. 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.
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