Topology of Configuration Space of Two Particles on a Graph, I

TL;DR

This paper investigates the homology and cohomology of the configuration space of two particles on a graph, providing explicit descriptions for planar graphs using intersection theory.

Contribution

It introduces a novel approach using intersection theory to analyze the cohomology algebra of configuration spaces on graphs, especially planar ones.

Findings

Explicit cohomology algebra description for planar graphs

Application of intersection theory to graph configuration spaces

Enhanced understanding of topological properties of particle configurations

Abstract

In this paper we study the homology and cohomology of confguration spaces of two distinct particles on a graph. Our main tool is intersection theory for cycles in graphs. We obtain an explicit description of the cohomology algebra of the configuration space in the case of planar graphs.