Section Problems for Graph Configuration Spaces

TL;DR

This paper investigates the conditions under which the natural projection maps between configuration spaces of a graph admit sections, analyzing how these conditions depend on the graph's homotopy type and providing construction techniques.

Contribution

It offers a complete characterization of when sections exist for configuration space maps based solely on the graph's homotopy type and introduces methods for constructing such sections.

Findings

Sections exist depending on the homotopy type of the graph.

Complete classification of graphs with sectionable configuration space maps.

Provides techniques for constructing sections.

Abstract

We consider, for a finite graph $G$, when the surjective map $\mathrm{Conf}_{n+1}(G) \rightarrow \mathrm{Conf}_n(G)$ of configuration spaces admits a section. We study when the answer depends only on the homotopy type of $G$, and give a complete answer. We also provide basic techniques for construction of sections.