Farley–Sabalka’s Morse-Theory Model and the Higher Topological Complexity of Ordered Configuration Spaces on Trees

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.