Relative Seiberg-Witten invariants and a sum formula

TL;DR

This paper introduces relative Seiberg-Witten invariants for 4-manifolds with embedded surfaces and establishes a sum formula relating invariants of a union to those of its parts, generalizing previous product formulas.

Contribution

It defines relative Seiberg-Witten invariants for pairs of 4-manifolds and surfaces, and proves a sum formula extending Morgan-Szabó-Taubes' product formula.

Findings

Defined relative Seiberg-Witten invariants for pairs (X,Σ).

Established a sum formula relating invariants of combined and component manifolds.

Generalized existing product formulas for SW invariants.

Abstract

We study relative Seiberg-Witten moduli spaces and define relative invariants for a pair $(X,\Sigma)$ consisting of a smooth, closed, oriented 4-manifold $X$ and a smooth, closed, oriented 2-dimensional submanifold $\Sigma\!\subset\!X$ with positive genus. These relative Seiberg-Witten invariants are meant to be the counterparts of relative Gromov-Witten invariants. We also obtain a sum formula (aka a product formula) that relates the SW invariants of a sum $X$ of two closed oriented 4-manifolds $X_1$ and $X_2$ along a common oriented surface $\Sigma$ with dual self-intersections to the relative SW invariants of $(X_1,\Sigma)$ and $(X_2,\Sigma)$. Our formula generalizes Morgan-Szab\'o-Taubes' product formula.