Singular maps on exotic 4-manifold pairs

TL;DR

This paper demonstrates that pairs of exotic 4-manifolds, which are topologically identical, can be distinguished by analyzing the topology of singular sets of smooth stable maps, utilizing Seiberg-Witten theory.

Contribution

It introduces a method to differentiate exotic 4-manifolds through the topology of singular sets in smooth maps, leveraging Seiberg-Witten invariants.

Findings

Exotic 4-manifold pairs can be distinguished by singular set topology.

Seiberg-Witten theory is effective in differentiating smooth structures.

Examples of such pairs are explicitly constructed.

Abstract

We show examples of pairs of smooth, compact, homeomorphic 4-manifolds, whose diffeomorphism types are distinguished by the topology of the singular sets of smooth stable maps defined on them. In this distinction we rely on results from Seiberg-Witten theory.