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Geometric and Functional Analysis

, Volume 24, Issue 4, pp 1080–1100 | Cite as

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

  • Gregory R. Chambers
  • Yevgeny Liokumovich
Article

Abstract

We prove two theorems about homotopies of curves on two-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.

Keywords

Orientable Surface Closed Curf Horizontal Edge Klein Bottle Simple Closed Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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