Topology of configuration space of two particles on a graph, II

TL;DR

This paper studies how adding edges to a graph affects the topology of the configuration space of two particles, introducing new algebraic tools and characterizing when the topology remains stable.

Contribution

It introduces a linking bilinear form on homology groups and provides explicit formulas for the homology of configuration spaces for mature graphs.

Findings

Explicit homology formulas for mature graphs

Edge addition preserves maturity under certain conditions

New algebraic tools for analyzing configuration space topology

Abstract

This paper continues the investigation of the configuration space of two distinct points on a graph. We analyze the process of adding an additional edge to the graph and the resulting changes in the topology of the configuration space. We introduce a linking bilinear form on the homology group of the graph with values in the cokernel of the intersection form (introduced in Part I of this work). For a large class of graphs, which we call mature graphs, we give explicit expressions for the homology groups of the configuration space. We show that under a simple condition, adding an edge to a mature graph yields another mature graph.