Abrams' stabilization theorem for no-k-equal configuration spaces on graphs

TL;DR

This paper extends Abrams' stabilization theorem for configuration spaces on graphs to include no-k-equal configurations, using discrete Morse theory to analyze the topological structure of these spaces.

Contribution

It generalizes Abrams' cubical complex model to no-k-equal configuration spaces on graphs, providing a new topological framework for these generalized spaces.

Findings

Established a deformation retract for no-k-equal configuration spaces

Applied discrete Morse theory to analyze the topology of these spaces

Extended the cubical complex model to a broader class of configuration spaces

Abstract

For a graph $G$, let Conf$(G,n)$ denote the classical configuration space of $n$ labelled points in $G$. Abrams introduced a cubical complex, denoted here by DConf$(G,n)$, sitting inside Conf$(G,n)$ as a strong deformation retract provided $G$ is suitably subdivided. Using discrete Morse Theory techniques, we extend Abrams' result to the realm of configurations having no $k$-fold collisions.