运动规划不变量与子群族
Motion planning invariants and families of subgroups
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中文总结 AI 辅助
本文引入关于子群族的拓扑复杂度不变量，提出置换族并证明其下界性质，从而推广了经典与分布拓扑复杂度相等的结果，并证明Farber猜想在分布意义上成立。
中文摘要 AI 辅助
我们引入了关于子群族的群拓扑复杂度的概念，其中平凡族和对角族的情形分别恢复了经典范畴和拓扑复杂度。这些不变量配备了一系列由函子性和Bredon上同调导出的上下界。我们引入了一个新的子群族，即置换族，并证明了相应不变量是Dranishnikov-Jauhari和Knudsen-Weinberger的分布拓扑复杂度的下界。作为第一个应用，我们证明了对一大类无挠群，置换族和对角族重合，从而经典拓扑复杂度和分布拓扑复杂度相等，推广了Dranishnikov的工作。其次，我们证明了Grant-Lupton-Oprea的下界实际上是分布拓扑复杂度的下界；特别地，由此得出Farber猜想在分布意义上成立。
英文摘要
We introduce a notion of topological complexity of a group with respect to a family of subgroups, with the case of the trivial and diagonal families recovering the classical category and topological complexity, respectively. These invariants come equipped with a battery of upper and lower bounds derived from functoriality and Bredon cohomology. We introduce a new family of subgroups, the permutational family, and show that the corresponding invariant is a lower bound for the distributional topological complexity of Dranishnikov-Jauhari and Knudsen-Weinberger. As a first application, we show that the permutational and diagonal families coincide, and thus that classical and distributional topological complexity are equal, for a large class of torsion-free groups, extending work of Dranishnikov. Second, we show that the lower bounds of Grant-Lupton-Oprea are in fact lower bounds on distributional topological complexity; in particular, it follows that Farber's conjecture holds distributionally.
发表机构
- University of Florida（佛罗里达大学）
- Colorado State University（科罗拉多州立大学）
机构由 AI 辅助整理，请以论文原文为准。
