On expansion and topological overlap
On expansion and topological overlap
复制标题
关于扩展和拓扑重叠
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Uli Wagner
中科院分区：
文献类型：
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作者：
Dominic Dotterrer;T. Kaufman;Uli Wagner
我们给出了格罗莫夫拓扑重叠定理的一个详细且易于理解的证明。设\(X\)是一个有限单纯复形，或者更一般地，是一个\(d\)维有限多面体胞腔复形。通俗地说，该定理表明，如果\(X\)具有足够强的高维扩张性质（它推广了图的边扩张，并且是根据\(X\)的胞腔上链来定义的），那么\(X\)具有以下拓扑重叠性质：对于每一个连续映射\(X
ightarrow\mathbb{R}^d\)，存在一个点\(p\in\mathbb{R}^d\)，它包含在\(X\)的\(d\) - 胞腔的正分数\(\mu>0\)的像中。更一般地，如果\(\mathbb{R}^d\)被任何\(d\)维分段线性流形\(M\)所取代，结论仍然成立，其中常数\(\mu\)仅取决于\(d\)和\(X\)的扩张性质，而不取决于\(M\)。
We give a detailed and easily accessible proof of Gromov’s Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X) then X has the following topological overlap property: for every continuous map $$X
ightarrow mathbb {R}^d$$X→Rd there exists a point $$pin mathbb {R}^d$$p∈Rd that is contained in the images of a positive fraction $$mu >0$$μ>0 of the d-cells of X. More generally, the conclusion holds if $$mathbb {R}^d$$Rd is replaced by any d-dimensional piecewise-linear manifold M, with a constant $$mu $$μ that depends only on d and on the expansion properties of X, but not on M.
DOI：
10.4171/owr/2016/46
发表时间：
2016
期刊：
Oberwolfach Reports
影响因子：
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作者：
Loeser F
通讯作者：
Loeser F