高维中的宏观标量曲率
Macroscopic Scalar Curvature in High Dimensions
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中文总结 AI 辅助
该研究描述了一种正宏观标量曲率版本，证明其可约束流形1-宽度，结合相关技术得到满足曲率条件的流形的有限覆叠同伦类型。
中文摘要 AI 辅助
我们描述了一种由Alpert、Balitskiy和Guth的工作启发的正宏观标量曲率版本，并证明流形满足该条件时，可根据其第一贝蒂数对其1-宽度给出界。证明中的关键工具是将任意闭流形分解为具有便利组合结构的链族，该分解受Nabutovsky、Rotman和Sabourau关于扫掠的工作启发。最后，利用Chodosh、Li和Liokumovich开发的技术，我们证明对于充分连通的流形，若其万有覆叠满足该曲率条件，则该流形存在有限覆叠，与$S^n$或$S^{n-1} \ imes S^1$的连通和同伦等价。
英文摘要
We describe a version of positive macroscopic scalar curvature motivated by the work of Alpert, Balitskiy, and Guth, and prove that this condition on a manifold implies a bound on its 1-width in terms of its first Betti number. A key tool in the proof is a decomposition of any closed manifold into a family of chains with convenient combinatorial structure, which was inspired by Nabutovsky, Rotman, and Sabourau's work on sweepouts. Finally, using techniques developed by Chodosh, Li, and Liokumovich, we show that for a sufficiently connected manifold, if the universal cover satisfies this curvature condition, then the manifold has a finite cover homotopy equivalent to $S^n$ or connected sums of $S^{n-1} \times S^1$.
