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Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: with an Appendix by Warren Dicks

About this Title

Joel Friedman, Department of Computer Science, University of British Columbia, Vancouver, BC V6T 1Z4, CANADA, and Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, CANADA.

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 233, Number 1100
ISBNs: 978-1-4704-0988-3 (print); 978-1-4704-1968-4 (online)
DOI: https://doi.org/10.1090/memo/1100
Published electronically: May 19, 2014
Keywords: Graphs, sheaves, Hanna Neumann Conjecture, homology, Galois theory., Intersection of free subgroups, graphs, skew group ring, sheaves, Hanna Neumann conjecture, homology, Galois theory.
MSC: Primary 20E05, 05C10, 55N30, 18F20; Secondary 20E08, 05C05, 05C50, 18F10, 18F20, 14F20

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Table of Contents

Chapters

• Preface
• Introduction
• 1. Foundations of Sheaves on Graphs and Their Homological Invariants
• 2. The Hanna Neumann Conjecture
• A. A Direct View of $\rho$-Kernels
• B. Joel Friedman’s Proof of the strengthened Hanna Neumann conjecture by Warren Dicks

Abstract

Abstract for Chapters 1,2, and Appendix A. 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.

Abstract for Appendix B. For a finite graph $Z$, $\bar {\mathrm {r}}(Z) := e-v+t$, where $e$, $v$, and $t$ denote the number of edges, vertices, and tree components of $Z$, respectively. Let $G$ be a finite group, $Z$ be a finite $G$-free $G$-graph, and $X$ and $Y$ be subgraphs of $Z$. Using linear algebra and algebraic geometry over a sufficiently large field, Joel Friedman proved that $\sum \limits _{g \in G} \bar {\mathrm {r}}(X\,{\cap }\,\,gY) \leqslant \bar {\mathrm {r}}(X) \bar {\mathrm {r}}(Y)$. He showed that this inequality implies the strengthened Hanna Neumann conjecture. We simplify Friedman’s proof of the foregoing inequality by replacing the sufficiently large field with a field $\mathbb {F}$ on which $G$ acts faithfully and then replacing all the arguments involving algebraic geometry with shorter arguments about the left ideals of the skew group ring $\mathbb {F}G$.