位形空间的横截性与“方桩”定理
Transversality for Configuration Spaces and the "Square-Peg" Theorem
摘要：
我们证明了一个transversality“提升性质”的紧致配置空间作为应用程序的multijet transversality定理：可以通过初始子流形的任意$C^1$ -小变化使欧几里得空间的任意子流形上的点的配置的子流形横截于欧几里得空间中的点的配置空间的任意子流形，只要紧化位形空间的两个子流形边界不相交。我们使用这种设置提供有吸引力的证据的存在的一些“特殊的内接配置”内家庭的领域嵌入在$\mathbb{R}^n$使用微分拓扑。例如，在平面上有一个光滑嵌入圆的$C^1$ -稠密族，其中每个简单闭曲线都有奇数个内接正方形，在$\mathbb{R}^n$中有一个光滑嵌入$(n-1)$ -球面的$C^1$ -稠密族，其中每个球面都有一个与$O(n)$同调的内接正则$n$ -单形族.
英文摘要：
We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclidean space by an arbitrarily $C^1$-small variation of the initial submanifold, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide attractive proofs of the existence of a number of "special inscribed configurations" inside families of spheres embedded in $\mathbb{R}^n$ using differential topology. For instance, there is a $C^1$-dense family of smooth embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a $C^1$-dense family of smooth embedded $(n-1)$-spheres in $\mathbb{R}^n$ where each sphere has a family of inscribed regular $n$-simplices with the homology of $O(n)$.
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