Constraints on embedded spheres and real projective planes in 4-manifolds from Seiberg-Witten theory

TL;DR

This paper uses Seiberg-Witten invariants to derive new constraints on how spheres and real projective planes can be embedded in 4-manifolds, revealing restrictions on their configurations and implications for manifold types.

Contribution

It introduces novel calculations of Seiberg-Witten invariants for branched covers and establishes new embedding constraints for spheres and projective planes in 4-manifolds.

Findings

New constraints on embedded spheres in 4-manifolds

Restrictions on embeddings of real projective planes

Implications for the existence of certain surface configurations

Abstract

We calculate the Seiberg-Witten invariants of branched covers of prime degree, where the branch locus consists of embedded spheres. Aside from the formula itself, our calculations give rise to some new constraints on configurations of embedded spheres in 4-manifolds. Using similar methods, we also obtain new constraints on embeddings of real projective planes and spheres with a cusp singularity. Moreover, we show that the existence of certain configurations of surfaces would give rise to 4-manifolds of non-simple type. Our proof makes use of equivariant Seiberg-Witten invariants as well as a gluing formula for the relative Seiberg-Witten invariants of 4-manifolds with positive scalar curvature boundary.