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

TL;DR

This paper studies fiber-connected, indefinite Morse 2-functions on connected manifolds, aiming to develop foundational tools for defining smooth invariants using Morse homology and Cerf theory.

Contribution

It introduces the concepts of fiber-connected and indefinite Morse 2-functions and explores their properties and homotopies, laying groundwork for future invariants.

Findings

Characterization of fiber-connected Morse 2-functions

Analysis of homotopies avoiding local extrema

Foundational framework for Morse homology on 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.