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- W4293578197 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." @default.
- W4293578197 created "2022-08-30" @default.
- W4293578197 creator A5038980736 @default.
- W4293578197 date "2011-04-30" @default.
- W4293578197 modified "2023-09-28" @default.
- W4293578197 title "Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture" @default.
- W4293578197 doi "https://doi.org/10.48550/arxiv.1105.0129" @default.
- W4293578197 hasPublicationYear "2011" @default.
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