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


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arn76.
    Arnold V.I.: Wave front evolution and equivariant Morse lemma. Communications on Pure and Applied Mathematics 29(6), 557–582 (1976)CrossRefMathSciNetGoogle Scholar
  2. Bae28.
    Baer R.: Isotopien von Kurven auf orientierbaren, geshlossenen Fächen. Journal für die Reine und Angewandte Mathematik 159, 101–116 (1928)zbMATHGoogle Scholar
  3. BM13.
    K. Burns and V. Matveev. Open problems and questions about geodesics (2013). Preprint, math arXiv:1308.5417.
  4. Bru86.
    Bruce J.W.: On transversality. Proceedings of the Edinburgh Mathematical Society 29, 115–123 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  5. CR13.
    G.R. Chambers and R. Rotman. Contracting loops on a Riemannian 2-surface (2013). Preprint, math arXiv:1311.2995 .
  6. Duf83.
    Dufour J.-P.: Familles de courbes planes différentiables. Topology 22(4), 449–474 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  7. Eps66.
    Epstein D.B.A.: Curves on 2-manifolds and isotopies. Acta Mathematica 115, 83–107 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  8. GG74.
    Golubitsky M., Guillemin V.: Stable Mappings and Their Singularities. Springer, New York (1974)Google Scholar
  9. HR13.
    Hingston N., Rademacher H.-B.: Resonance for loop homology of spheres. The Journal of Differential Geometry 93(1), 133–174 (2013)zbMATHMathSciNetGoogle Scholar
  10. Hir94.
    Hirsch M.W.: Differential Topology. Springer, New York (1994)Google Scholar
  11. Mur96.
    Murasugi K.: Knot Theory and its Applications. Birkhäuser, Boston (1996)zbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations