Sheaves on Graphs and Their Homological Invariants

TL;DR

This paper develops a sheaf-theoretic framework on graphs, introducing the maximum excess invariant, which generalizes classical graph invariants and has properties akin to Betti numbers and L^2 Betti numbers.

Contribution

It introduces a novel sheaf theory on graphs with the maximum excess invariant, connecting homological invariants to classical graph theory and expanding algebraic graph theory.

Findings

Maximum excess is a supermodular, limit-like invariant.

Maximum excess relates to Betti numbers and graph expansion.

Framework generalizes algebraic graph theory concepts.

Abstract

We introduce a notion of a sheaf of vector spaces on a graph, and develop the foundations of homology theories for such sheaves. One sheaf invariant, its "maximum excess," has a number of remarkable properties. It has a simple definition, with no reference to homology theory, that resembles graph expansion. Yet it is a "limit" of Betti numbers, and hence has a short/long exact sequence theory and resembles the $L^2$ Betti numbers of Atiyah. Also, the maximum excess is defined via a supermodular function, which gives the maximum excess much stronger properties than one has of a typical Betti number. The maximum excess gives a simple interpretation of an important graph invariant, which will be used to study the Hanna Neumann Conjecture in a future paper. Our sheaf theory can be viewed as a vast generalization of algebraic graph theory: each sheaf has invariants associated to it---such as Betti numbers and Laplacian matrices---that generalize those in classical graph theory.