A Proof of the Strengthened Hanna Neumann Conjecture

TL;DR

This paper proves the Strengthened Hanna Neumann Conjecture using a novel cohomological approach involving graph Galois theory and sheaf kernels, providing a new perspective on the conjecture's structure.

Contribution

It introduces a cohomological interpretation of the conjecture and employs sheaf theory and Galois theory to prove a generalized version, which was previously unapproachable.

Findings

Proof of the Strengthened Hanna Neumann Conjecture.

Introduction of a cohomological framework for the conjecture.

Identification of kernels that are co-acyclic in covering cohomology.

Abstract

We prove the Strengthened Hanna Neumann Conjecture. We give a more direct cohomological interpretation of the conjecture in terms of "typical" covering maps, and use graph Galois theory to "symmetrize" the conjecture. The conjecture is then related to certain kernel of a morphism of sheaves, and is implied provided these kernels are co-acyclic in the covering cohomology theory. This allows us to prove a slightly generalized Strengthened Hanna Neumann Conjecture; this conjecture is false if generalized to all sheaves. The kernels we use do not exist in the theory of graphs, so our use of sheaf theory seems essential to this approach.