欧几里得空间粗分离的有限方法
A Finitary Approach to Coarse Separation of Euclidean spaces
AI总结：
用奇异同调/上同调及代数拓扑标准事实，将欧几里得空间粗分离子集族问题简化为有限版本，证其渐近维数至少为d - 1，还得出相关空间结论及对粗嵌入的阻碍。
摘要：
我们给出一个新证明：欧几里得空间\(\mathbb{R}^{d}\)的每个粗分离子集族必定有至少为\(d - 1\)的渐近维数。证明仅使用奇异同调/上同调及代数拓扑的标准事实，如亚历山大对偶性。通过先将问题简化为其有限版本来做到这一点。利用我们的方法，立即可得\(d\)维欧几里得建筑或\(d\)个测地线、测地完备度量空间乘积的每个粗分离子集族有至少为\(d - 1\)的渐近维数。作为推论，我们得到欧几里得空间到某些群图的基本群的粗嵌入的阻碍。
英文摘要：
We give a novel proof of the fact that every coarsely separating family of subsets of the Euclidean space $\mathbb{R}^{d}$ must have asymptotic dimension at least $d-1$. The proof only uses singular homology/cohomology and standard facts from algebraic topology, such as Alexander duality. We do this by first reducing the problem to a finitary version of it. Using our approach, it follows immediately that every coarsely separating family of subsets of a $d$-dimensional Euclidean building or a product of $d$ geodesic, geodesically complete metric spaces has asymptotic dimension at least $d-1$. As a corollary, we obtain obstructions to coarse embeddings of Euclidean spaces into certain fundamental groups of graphs of groups.
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。
