Mathematics Colloquia and Seminars

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Work of various people \cite{Gay-Kirby, Constructing Lefschetz-type fibrations on four-manifolds Geometry & Topology 11 (2007) 2075--2115 and Lekili, Wrinkled fibrations on near-symplectic manifolds, Geometry & Topology 13 (2009) 277--318} has shown that any smooth, orientable 4-manifold, X, is a broken fibration over S^2 when X has no boundary, or over B^2 otherwise. Such a map is a fibration except along smooth circles where round 1-handles are attached. Such a map is also locally complex away from the circles, that is, it has rank 2 or 0. More recent work (Gay-Kirby, Jonathan Williams, Douglas-Gualitieri) shows that natural moves suffice to go from one broken fibration to another on the same 4-manifold. These issue will be discussed over three lectures, beginning with elementary material on maps to the reals (Morse theory and Cerf theory) followed by maps to R^2 and a bit more singularity. This is then interpreted in the case of maps from 4-manifolds to 2-manifolds, and in terms of handle theory.