4:00 pm Thursday, October 10, 2013

David Gay (University of Georgia)

This is joint work with R. Kirby. I will show how to chop up any closed, connected, oriented 4-manifold into 3 diffeomorphic pieces, each diffeomorphic to a thickening in R^4 of a graph, fitting together in a particularly nice way. This is motivated by the fact that 3-manifolds can be split into 2 such pieces (graphs thickened in R^3); such splittings are called Heegaard splittings and are tremendously useful in understanding 3-manifolds. One of the main points is that this result is simple, the proof is not that hard, and the parallels with the 3-dimensional results are striking. We also have a uniqueness result, uniqueness up to a natural stabilization operation, that also precisely parallels the 3-dimensional version, the Reidemeister-Singer theorem. Along the way I will present lots of natural open questions.