An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs
Emilio J González, Jesús González · Algebraic & Geometric Topology · 2024
We introduce and study an algorithm that constructs a discrete gradient field on any simplicial complex.With a computational complexity similar to that of existing methods, our algorithmic gradient field is always maximal and in a number of cases even optimal.We make a thorough analysis of the resulting gradient field in the case of Munkres discrete model for Conf.K m ; 2/, the configuration space of ordered pairs of noncolliding particles moving on the complete graph K m on m vertices.This allows us to describe in full the cohomology algebra H .Conf.K m ; 2/I R/ for any commutative unital ring R. As an application we prove that, although Conf.K m ; 2/ is outside the "stable" regime, all its topological complexities are maximal when m 4.