论文标题
完整图上两个点的配置空间的算法离散梯度字段和共同体学代数
An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs
论文作者
论文摘要
我们介绍了一种算法，该算法在任何简单复合物上构建一个离散的梯度场。我们表明，在所有情况下，梯度场都是最大的，并且在许多情况下，最佳。如果Munkres的离散模型$ \ text {conf}（k_m，2）$，我们对所得梯度字段进行了彻底的分析，这是$ m $ $ Vertices上的完整图$ k_m $上有序的非收集粒子的配置空间。加上Forman的离散摩尔斯理论，这使我们能够全面描述$ r $ r $ -algebra $ h^*（\ text {conf}（k_m，2）; r）$，用于任何交换Unitial Ring $ r $。作为一个应用程序，我们证明，尽管$ \ text {conf}（k_m，2）$在“稳定”制度之外，但当$ m \ geq4 $时，其所有拓扑复杂性都是最大的。
We introduce an algorithm that constructs a discrete gradient field on any simplicial complex. We show that, in all situations, the gradient field is maximal possible and, in a number of cases, optimal. We make a thorough analysis of the resulting gradient field in the case of Munkres' discrete model for $\text{Conf}(K_m,2)$, the configuration space of ordered pairs of non-colliding particles on the complete graph $K_m$ on $m$ vertices. Together with the use of Forman's discrete Morse theory, this allows us to describe in full the cohomology $R$-algebra $H^*(\text{Conf}(K_m,2);R)$ for any commutative unital ring $R$. As an application we prove that, although $\text{Conf}(K_m,2)$ is outside the "stable" regime, all its topological complexities are maximal possible when $m\geq4$.
