Constraints for Seiberg-Witten basic classes of glued manifolds

TL;DR

This paper explores how Seiberg-Witten basic classes behave under the operation of gluing 4-manifolds along Riemann surfaces, providing conditions and examples for their complete description.

Contribution

It introduces constraints on Seiberg-Witten basic classes for glued 4-manifolds using gluing theory, advancing understanding of their behavior in complex topological constructions.

Findings

Derived relations between basic classes of component and glued manifolds

Identified cases where basic classes can be completely described

Provided examples illustrating the theoretical results

Abstract

We use rudiments of the Seiberg-Witten gluing theory for trivial circle bundles over a Riemann surface to relate de Seiberg-Witten basic classes of two $4$-manifolds containing Riemann surfaces of the same genus and self-intersection zero with those of the $4$-manifold resulting as a connected sum along the surface. We study examples in which this is enough to describe completely the basic classes.