Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture

Understand

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.