Graph homology and graph configuration spaces

TL;DR

This paper explores the homology of graph configuration spaces using spectral sequences, revealing how graph cohomology and Massey products determine the homological structure.

Contribution

It establishes a connection between the spectral sequence's E_1 term and graph cohomology, extending previous conjectures to arbitrary graphs and DG algebras.

Findings

E_1 term given by graph cohomology complex C_A(G)

Higher differentials derived from Massey products of A

Results apply to arbitrary finite graphs and DG algebras

Abstract

If $R$ is a commutative ring, $M$ a compact $R$-oriented manifold and $G$ a finite graph without loops or multiple edges, we consider the graph configuration space $M^G$ and a Bendersky-Gitler type spectral sequence converging to the homology $H_*(M^G, R)$. We show that its $E_1$ term is given by the graph cohomology complex $C_A(G)$ of the graded commutative algebra $A = H^*(M, R)$ and its higher differentials are obtained from the Massey products of $A$, as conjectured by Bendersky and Gitler for the case of a complete graph $G$. Similar results apply to the spectral sequence constructed from an arbitrary finite graph $G$ and a graded commutative DG algebra $\mathcal{A}$.