论文标题
通过Urysohn宽度测量的地图的腰部
Waist of maps measured via Urysohn width
论文作者
论文摘要
我们讨论了以下各种问题：对于连续地图$ x \ y $从紧凑的度量空间到简单综合体，可以保证在Urysohn宽度的意义上存在较大的纤维吗？ $ d $ - 宽度可以衡量一个$ d $维的综合体可以近似空间的能力。本文的结果包括以下内容。 1) Any piecewise linear map $f: [0,1]^{m+2} \to Y^m$ from the unit euclidean $(m+2)$-cube to an $m$-polyhedron must have a fiber of $1$-width at least $\frac{1}{2βm +m^2 + m + 1}$, where $β= \sup_y \ text {rk} h_1（f^{ - 1}（y））$测量地图的拓扑复杂性。 2）存在一个分段平滑的地图$ x^{3m+1} \ to \ mathbb {r}^m $，带有$ x $ a riemannian $（3m+1）$ - 大300万美元$ -WIDTH的分歧，所有Fibers均为拓扑$（2M+1）$（2M+1）$ - $ - $ 1 $ 1）$（M+1）$（M+1）。
We discuss various questions of the following kind: for a continuous map $X \to Y$ from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The $d$-width measures how well a space can be approximated by a $d$-dimensional complex. The results of this paper include the following. 1) Any piecewise linear map $f: [0,1]^{m+2} \to Y^m$ from the unit euclidean $(m+2)$-cube to an $m$-polyhedron must have a fiber of $1$-width at least $\frac{1}{2βm +m^2 + m + 1}$, where $β= \sup_y \text{ rk } H_1(f^{-1}(y))$ measures the topological complexity of the map. 2) There exists a piecewise smooth map $X^{3m+1} \to \mathbb{R}^m$, with $X$ a riemannian $(3m+1)$-manifold of large $3m$-width, and with all fibers being topological $(2m+1)$-balls of arbitrarily small $(m+1)$-width.
