Topology Seminar: Joel Friedman (UBC)

In these talks we describe our proofs of the Hanna Neumann Conjecture. This conjecture of the late 1950's can be described both as a problem in group theory or as one in graph theory. Our first proof is longer and interprets the problem using homology of sheaves on graphs; our second proof is very short and uses only simple graph theory, but was inspired from the type of induction used in our first proof. Both proofs demonstrate the strengthened form of the conjecture formulated by Walter Neumann, and both proofs use earlier resolved cases of the conjecture.

A crucial idea of the proof is to express the seemingly awkward notion of "reduced cyclicity" of a graph in simpler terms, involving limits over covering maps. This "limit homology theory" may be of independent interest, and is related to the Atiyah Conjecture; our theory requires some curious linear algebra that also may be of independent interest.

Our talks will not assume any previous knowledge of sheaf theory.