Converting homotopies to isotopies and dividing homotopies in half in an effective way
Converting homotopies to isotopies and dividing homotopies in half in an effective way
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有效地将同伦转化为同伦并将同伦一分为二
DOI：
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发表时间：
2013
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通讯作者：
Yevgeny Liokumovich
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文献类型：
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作者：
Gregory R. Chambers;Yevgeny Liokumovich
我们证明了二维黎曼流形上曲线同伦的两个定理。我们证明，对于任何 $${\epsilon > 0}$$ϵ>0，如果两条简单闭合曲线通过有界长度 L 的曲线是同伦的，那么它们也是通过以 $${L + \epsilon}$$L+ϵ 为界的长度曲线同位的。如果流形是可定向的，那么对于任何$${\epsilon > 0}$$ϵ>0，我们表明，如果我们可以收缩一条通过长度为L的曲线两次遍历的曲线$${\gamma}$$γ，那么我们也可以通过长度为$${L + \epsilon}$$L+ϵ为界的曲线收缩$${\gamma}$$γ。我们的方法包括在自交点处切割曲线并以规定的方式重新连接它们。我们考虑以这种方式从原始同伦获得的所有曲线的空间，并使用一种新颖的方法来表明该空间包含产生所需同伦的路径。
We prove two theorems about homotopies of curves on two-dimensional Riemannian manifolds. We show that, for any $${\epsilon > 0}$$ϵ>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 $${L + \epsilon}$$L+ϵ. If the manifold is orientable, then for any $${\epsilon > 0}$$ϵ>0 we show that, if we can contract a curve $${\gamma}$$γ traversed twice through curves of length bounded by L, then we can also contract $${\gamma}$$γ through curves bounded in length by $${L + \epsilon}$$L+ϵ. 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.