Farley-Sabalka's Morse-theory model and the higher topological complexity of ordered configuration spaces on trees

TL;DR

This paper develops a Morse-theory model for ordered configuration spaces on trees, enabling the calculation of higher topological complexities and generalizing previous homotopy descriptions for two particles.

Contribution

It introduces a Morse-theory framework based on a discrete gradient field to analyze the topology of configuration spaces on trees, extending existing results to any number of particles.

Findings

Computed all higher topological complexities for ordered configuration spaces on trees.

Generalized homotopy type descriptions from two particles to any number of particles.

Provided a new Morse-theoretic approach to study configuration spaces on graphs.

Abstract

Using the ordered analogue of Farley-Sabalka's discrete gradient field on the configuration space of a graph, we unravel a levelwise behavior of the generators of the pure braid group on a tree. This allows us to generalize Farber's equivariant description of the homotopy type of the configuration space on a tree on two particles. The results are applied to the calculation of all the higher topological complexities of ordered configuration spaces on trees on any number of particles.