# Mathematical Sciences Research Institute

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# Seminar

Research Seminar: "Indefinite Morse 2-functions" April 13, 2010
Parent Program: -- MSRI: Simons Auditorium
Speaker(s) David Gay
Description No Description
Video
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