A Proof of the Strengthened Hanna Neumann Conjecture

May 15, 2010: This paper has been corrected and is being split into two smaller papers. The first paper, The Strengthened Hanna Neumann Conjecture I: A Combinatorial Proof, is now available. In this paper the sheaf theory has been written in purely combinatorial terms. A second paper is in preparation, to describe the sheaf theory and give a very short proof of the conjecture ("very short" assuming known cases of the conjecture and assuming the sheaf theory is in place).
June 12, 2009: An error was found in the paper; we are trying to fix the error and post an updated manuscript before we distribute the paper further.
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  • 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.