0905.3781May 23, 2009This paper was been withdrawn by the author due to a crucial error in thinking that it is immediate that the vanishing of rho kernels of a direct sum of sheaves is equivalent to the individual vanishing. A corrected version of the proof is available as "The Strengthened Hanna Neumann Conjecture I", which gives a proof without sheaf theory (really translating the sheaf theory to combinatorics). A sequel to this paper, "The Strengthened Hanna Neumann Conjecture II" is in preparation; this will develop sheaf theory and give a simple proof of the conjecture (simple assuming the sheaf theory is in place).
A Proof of the Strengthened Hanna Neumann Conjecture
Joel Friedman
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.