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arXiv · 2207.07046

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

Emilio J. González·Jesús González
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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 $H^*(\text{Conf}(K_m,2);R)$ 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 $m\geq4$.

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Emilio J. González, Jesús González. 2022-07-14. An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs. https://doi.org/10.2140/agt.2024.24.3719

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