# Configuration spaces of graphs with certain permitted collisions

## TL;DR

This paper studies configuration spaces of points on graphs where collisions are permitted only at vertices, computing their fundamental and homology groups for trees and establishing stability properties for general graphs.

## Contribution

It introduces and analyzes a new class of configuration spaces on graphs with vertex-only collisions, providing homological computations and stability results.

## Key findings

- Fundamental groups and homology of configuration spaces on trees are computed.
- Homology groups of these spaces exhibit generalized representation stability.
- Results extend to general graphs for large numbers of points.

## Abstract

If $G$ is a graph with vertex set $V$, let Conf$_n^{\text{sink}}(G,V)$ be the space of $n$-tuples of points on $G$, which are only allowed to overlap on elements of $V$. We think of Conf$_n^{\text{sink}}(G,V)$ as a configuration space of points on $G$, where points are allowed to collide on vertices. In this paper, we attempt to understand these spaces from two separate, but closely related, perspectives. Using techniques of combinatorial topology we compute the fundamental groups and homology groups of Conf$_n^{\text{sink}}(G,V)$ in the case where $G$ is a tree. Next, we use techniques of asymptotic algebra to prove statements about Conf$_n^{\text{sink}}(G,V)$, for general graphs $G$, whenever $n$ is sufficiently large. It is proven that, for general graphs, the homology groups exhibit generalized representation stability in the sense of previous work of the author.