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Optimal homotopies of curves on surfaces

Banff International Research Station for Mathematical Innovation and Discovery

In this talk, we will prove the following theorems. For any ǫ > 0, we have that:\\r\\n(1) If two simple closed curves on a 2-dimensional Riemannian manifold are homotopic through loops of length at most L, then they are also homotopic through simple closed curves of length at most L + ǫ (joint work with Y. Liokumovich).\\r\\n(2) If the boundary of a Riemannian 2-disc can be contracted through closed curves of length at most L, then it can be contracted through based loops of length at most L + 2D + ǫ, where D is the diameter of the 2-disc (joint work with R. Rotman). This result can be generalized for simple closed curves on Riemannian 2-manifolds.\\r\\n(3) A closed curve on an orientable Riemannian 2-manifold can be con- tracted through loops of length at most L + ǫ if the curve formed by traversing twice can be contracted through loops of length at most L (joint work with Y. Liokumovich). This can be seen as a quantitative version of the fact that the fundamental group of an orientable surface contains no elements of order 2.

Mathematics; Differential geometry; Manifolds and cell complexes

video/mp4

Author affiliation: University of Toronto

2014-08-06

Attribution-NonCommercial-NoDerivs 2.5 Canada

http://hdl.handle.net/2429/49425

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/2.5/ca/