Stirling Decomposition of Graph Homology in Genus 1

TL;DR

This paper establishes a decomposition of genus 1 commutative graph homology into summands with ranks given by Stirling numbers, computed via homology of decorated tree complexes, with an elementary combinatorial approach.

Contribution

It introduces a novel direct sum decomposition of genus 1 graph homology using Stirling numbers and decorated tree complexes, providing an elementary combinatorial perspective.

Findings

Homology ranks are given by Stirling numbers of the first kind.

Decomposition applies to genus 1 with at least 3 markings.

Decorated tree complexes are used to compute homology.

Abstract

We prove that commutative graph homology in genus $g=1$ with $n\geq 3$ markings has a direct sum decomposition whose summands have rank given by Stirling numbers of the first kind. These summands are computed as the homology of complexes of certain decorated trees. This paper was written with a non-expert audience in mind, and an emphasis is placed on an elementary combinatorial description of these decorated tree complexes.