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

TL;DR

This paper introduces a new proof for computing cobordism groups of Morse functions on surfaces and 3-manifolds, utilizing universal complexes of singular fibers to derive invariants and apply to map germs.

Contribution

It provides a simplified method for calculating cobordism groups and introduces universal complexes as complete invariants, with applications to map germs and cusp counting.

Findings

Computed cobordism groups for Morse functions on surfaces and 3-manifolds.

Identified cohomology classes as complete invariants for these cobordism groups.

Demonstrated that cusp counts in stable perturbations are topological invariants.

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: (R^3, 0) --> (R^2, 0) with R^2 being oriented, the algebraic number of cusps appearing in a stable perturbation of g is a local topological invariant of g.