|Location:||MSRI: Simons Auditorium|
This is joint work with Rob Kirby, inspired and motivated by work of Tim Perutz and Yanki Lekili on the possibility of extracting
smooth 4-manifold invariants from broken Lefschetz fibrations over the sphere. A "Morse 2-function" is a suitably generic smooth map from an n-manifold to a 2-manifold, just as a Morse function is a suitably generic map to a 1-manifold. Locally, Morse 2-functions look like $(t,p) \mapsto (t,g_t(p))$, where $g_t$ is a generic homotopy between Morse functions (on an (n-1)--manifold), so thinking about Morse 2-functions is something like thinking about Cerf theory when you can't say globally what direction should be called "time". An indefinite Morse 2-function is one in which, in this local model, the Morse function $g_t$ never has critical points of minimal or maximal index. We prove existence and uniqueness results for indefinite Morse 2-functions over the disk and the sphere. "Uniqueness" means that homotopic indefinite Morse 2-functions can be connected by generic
homotopies which are indefinite at all intermediate times. Some of our results have already been proved by Saeki and Williams, but we have a number of important improvements, most notably that we can keep fibers connected at all times and that the results for the sphere follow as corollaries from the results for the disk. We also think that the perspective on Morse 2-functions which comes from our methods of proof is enlightening in and of itself and worth sharing.