Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture

Abstract

In this paper we establish some foundations regarding sheaves of vector spaces on graphs and their invariants, such as homology groups and their limits. We then use these ideas to prove the Hanna Neumann Conjecture of the 1950's; in fact, we prove a strengthened form of the conjecture. 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. 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. We shall use "Galois graph theory" to reduce the Strengthened Hanna Neumann Conjecture to showing that certain sheaves, that we call $\rho$-kernels, have zero maximum excess. We use the symmetry in Galois theory to argue that if the Strengthened Hanna Neumann Conjecture is false, then the maximum excess of "most of" these $\rho$-kernels must be large. We then give an inductive argument to show that this is impossible.

A previous version of this paper was merged with a another paper, arXiv:1104.2265. Redundancies between the two earlier papers have been eliminated, and some comments were added

References in corpus (3)

• A short proof that positive generation implies the Hanna Neumann Conjecture
• On the Hanna Neumann Conjecture
• Sheaves on Graphs and Their Homological Invariants

Cited by in corpus (5)

• The Relativized Second Eigenvalue Conjecture of Alon
• On the intersections of finitely generated subgroups of free groups: reduced rank to full rank
• A note on the finitely generated fixed subgroup property
• The Hanna Neumann Conjecture for graphs of free groups with cyclic edge groups
• An analogue of the strengthened Hanna Neumann conjecture for virtually free groups and virtually free products