The Homology of Configuration Spaces of Trees with Loops

TL;DR

This paper proves that the homology of ordered configuration spaces of finite trees with loops is torsion free, introduces configuration spaces with sinks, and provides explicit generators for their homology groups.

Contribution

It introduces configuration spaces with sinks, enabling quotients, and offers a concrete generating set for homology groups of these spaces, advancing understanding of their algebraic topology.

Findings

Homology of configuration spaces of trees with loops is torsion free.

Introduction of configuration spaces with sinks for quotients.

Explicit generators for homology groups of these configuration spaces.

Abstract

We show that the homology of ordered configuration spaces of finite trees with loops is torsion free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete generating set for all homology groups of configuration spaces of trees with loops and the first homology group of configuration spaces of general finite graphs. An important technique in the paper is the identification of the $E^1$-page and differentials of Mayer-Vietoris spectral sequences for configuration spaces.