An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs

Abstract

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 $H^{\ast }(\text{Conf}(K_m,2);R)$ 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 $m\ge 4$.

29 pages, 18 figures

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• Higher topological complexity and its symmetrization
• Presentations of Graph Braid Groups
• The Homology of Configuration Spaces of Trees with Loops
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• Topology of configuration space of two particles on a graph, II