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

TL;DR

This paper presents an algorithm for constructing optimal discrete gradient fields on simplicial complexes, enabling a complete description of the cohomology algebra of configuration spaces of two points on complete graphs, with implications for their topological complexity.

Contribution

It introduces a new algorithm for maximal discrete gradient fields and applies it to fully determine the cohomology algebra of configuration spaces on complete graphs.

Findings

The gradient field constructed is maximal and sometimes optimal.

The cohomology algebra of configuration spaces on complete graphs is fully described.

Topological complexities are maximal for these spaces when m ≥ 4.

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 $m\geq4$.