Homology groups for particles on one-connected graphs

TL;DR

This paper develops a mathematical framework to compute homology groups of configuration spaces for particles on one-connected graphs, aiding the understanding of quantum statistics in such systems.

Contribution

It introduces a new approach combining combinatorial properties, Mayer-Vietoris sequences, and discrete Morse theory to analyze the topology of configuration spaces on graphs.

Findings

Derived closed-form formulas for homology group ranks on tree graphs.

Computed the second homology group for distinguishable and indistinguishable particles.

Provided insights into the topology relevant for quantum statistics.

Abstract

We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our approach is based on some fundamental combinatorial properties of the configuration spaces, Mayer-Vietoris sequences for different parts of configuration spaces and some limited use of discrete Morse theory. As one of the results, we derive a closed-form formulae for ranks of the homology groups for indistinguishable particles on tree graphs. We also give a detailed discussion of the second homology group of the configuration space of both distinguishable and indistinguishable particles. Our motivation is the search for new kinds of quantum statistics.