Configuration Graph Cohomology

TL;DR

This paper introduces a new cohomology-based approach to analyze the fundamental groups of configuration spaces on surfaces, where points are grouped by colors and their collision rules are dictated by a graph.

Contribution

It reformulates the problem of describing solutions to linear Diophantine equations using novel graph cohomology groups, linking solutions to graph properties.

Findings

Solution sets depend on graph properties.

Cohomology groups encode collision constraints.

Results relate algebraic solutions to topological features.

Abstract

We investigate an algebraic problem related to the determination of the fundamental group of a class of spaces of configurations on surfaces. The configuration spaces are spaces of points grouped into colors. Whether two points are allowed to collide is determined by a graph, whose vertices are the colors. In an earlier paper, the fundamental group of such graphs was described as solutions to linear Diophantine equations. In this paper, the problem of describing the set of solution is reformulated using a new type of cohomology groups of graphs. The dependence of the solution on the number of points of each color is studied. The answer is formulated in terms of graph theoretical properties.