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We discuss generic smooth maps from smooth manifolds to smooth surfaces,which we call "Morse 2-functions", and homotopies between such maps. The twocentral issues are to keep the fibers connected, in which case the Morse2-function is "fiber-connected", and to avoid local extrema over 1-dimensionalsubmanifolds of the range, in which case the Morse 2-function is "indefinite".This is foundational work for the long-range goal of defining smooth invariantsfrom Morse 2-functions using tools analogous to classical Morse homology andCerf theory.

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