Singular fibers of stable maps and signatures of 4-manifolds

TL;DR

This paper establishes a direct relationship between the algebraic count of certain singular fibers in stable maps from 4-manifolds to 3-manifolds and the signature of the 4-manifold, revealing a new topological invariant connection.

Contribution

It proves that the algebraic number of specific singular fibers in stable maps equals the signature of the 4-manifold, linking singularity theory with 4-manifold topology.

Findings

Algebraic count of singular fibers equals the manifold's signature

Establishes a new invariant relationship in 4-manifold topology

Provides a method to compute signatures via stable maps

Abstract

We show that for a C^infty stable map of an oriented 4-manifold into a 3-manifold, the algebraic number of singular fibers of a specific type coincides with the signature of the source 4-manifold.