Singular fibers of stable maps of 3-manifolds with boundary into surfaces and their applications

TL;DR

This paper classifies singular fibers of stable maps from 3-manifolds with boundary into surfaces, computes related cohomology groups, and derives cobordism invariants for Morse functions on surfaces with boundary.

Contribution

It provides a comprehensive classification of singular fibers for these maps and introduces cobordism invariants for Morse functions on bordered surfaces.

Findings

Classification of singular fibers for proper stable maps from 3-manifolds with boundary

Computation of cohomology groups of the universal complex of singular fibers

Derivation of cobordism invariants for Morse functions on surfaces with boundary

Abstract

In this paper, we first classify singular fibers of proper $C^\infty$ stable maps of 3-dimensional manifolds with boundary into surfaces. Then, we compute the cohomology groups of the associated universal complex of singular fibers, and obtain certain cobordism invariants for Morse functions on compact surfaces with boundary.