Fiber connected, indefinite Morse 2-functions on connected n-manifolds

Abstract

We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of the range, in which case the Morse 2-function is "indefinite". This is foundational work for the long-range goal of defining smooth invariants from Morse 2-functions using tools analogous to classical Morse homology and Cerf theory.

12 pages, 8 figures, to appear in Proc. Nat. Acad. Sci. U.S.A. This contains a condensed presentation of most of the results in arXiv:1102.0750 as well as some alternative perspectives. Revised to add a reference

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