配置空间、多重射流横截性与方钉问题
Configuration Spaces, Multijet Transversality, and the Square-Peg Problem
AI总结：
本文利用多重射流横截性定理证明紧化配置空间的横截提升性质，并由此给出方钉问题的新证明：存在稠密光滑嵌入圆族，使每条简单闭曲线具有奇数个内接正方形或正方形类四边形。
摘要：
我们证明了紧化配置空间的一个横截性“提升性质”，作为多重射流横截性定理的一个应用：给定欧氏空间中紧流形 $M$ 的一个嵌入上点配置的一个子流形，只要紧化配置空间的两个子流形边界不相交，我们就能找到 $M$ 的一族稠密光滑嵌入，使得相应的点配置空间与欧氏空间中点配置空间的任意子流形横截。我们利用这一框架给出了方钉问题的一个有吸引力的证明：平面中存在一族稠密的光滑嵌入圆，其中每条简单闭曲线都有奇数个内接正方形；并且在 $\mathbb{R}^n$ 中存在一族稠密的光滑嵌入圆，其中每条简单闭曲线都有奇数个内接正方形类四边形。
英文摘要：
We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: given a submanifold of configurations of points on an embedding of a compact manifold $M$ in Euclidean space, we can find a dense set of smooth embeddings of $M$ for which the corresponding configuration space of points is transverse to any submanifold of the configuration space of points in Euclidean space, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide an attractive proof of the square-peg problem: there is a dense family of smoothly embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a dense family of smoothly embedded circles in $\mathbb{R}^n$ where each simple closed curve has an odd number of inscribed square-like quadrilaterals.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
