Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs

Abstract

We give a new and simple proof for the computation of the oriented and the unoriented fold cobordism groups of Morse functions on surfaces. We also compute similar cobordism groups of Morse functions based on simple stable maps of $3$–manifolds into the plane. Furthermore, we show that certain cohomology classes associated with the universal complexes of singular fibers give complete invariants for all these cobordism groups. We also discuss invariants derived from hypercohomologies of the universal homology complexes of singular fibers. Finally, as an application of the theory of universal complexes of singular fibers, we show that for generic smooth map germs $g:\left({ℝ}^3,0\right)\to \left({ℝ}^2,0\right)$ with ${ℝ}^2$ being oriented, the algebraic number of cusps appearing in a stable perturbation of $g$ is a local topological invariant of $g$.