论文标题
运动计划和拓扑复杂性有限的概括
A Finite Equivariant Generalization of Motion Planning and Topological Complexity
论文作者
论文摘要
本文探讨了有限的模棱两可环境中的拓扑复杂性。我们首先定义并研究了有限拓扑空间的田中组合复杂性的模棱两可的版本。我们探讨了这种不变的与文献中已经讨论过的其他几个不变的关系：Farber的拓扑复杂性，田中的组合复杂性以及Colman-Grant的Equivariant Lusternik-Schnirelmann类别。我们找到了均衡的组合复杂性和等效组合运动策略的必要长度的界限。我们表明，任何有限的$ g $空间的拓扑复杂性都等于其模棱两可的组合复杂性。 然后，我们将Gonzàlez的简单复杂性调整为订购和无序的$ G $ - 简化复合物，并探索其第一个属性。最后，我们表明，任何有序的$ g $ simplicial复合物的实现的拓扑复杂性等于等效的简单复杂性。
This paper explores topological complexity in the finite equivariant setting. We first define and study an equivariant version of Tanaka's combinatorial complexity for finite topological spaces. We explore the relationships between this invariant and several others already discussed in the literature: Farber's topological complexity, Tanaka's combinatorial complexity, and Colman-Grant's equivariant Lusternik-Schnirelmann category. We find bounds for equivariant combinatorial complexity and for the necessary lengths of equivariant combinatorial motion plannings. We show that the equivariant topological complexity of any finite $G$-space is equal to its equivariant combinatorial complexity. We then adapt Gonzàlez's simplicial complexity to ordered and unordered $G$-simplicial complexes and explore its first properties. Lastly, we show that the equivariant topological complexity of the realization of any ordered $G$-simplicial complex is equal to the equivariant simplicial complexity.
