# A cohomological Seiberg-Witten invariant emerging from the adjunction inequality

## TL;DR

This paper introduces a new cohomological Seiberg-Witten invariant for 4-manifolds, derived from families of equations, providing novel genus constraints on embedded surfaces, especially in cases where traditional invariants vanish.

## Contribution

It constructs a cohomological invariant from families of Seiberg-Witten equations, offering new insights into the topology of 4-manifolds and embedded surfaces.

## Key findings

- Invariant is non-zero for certain positive scalar curvature 4-manifolds.
- Provides new adjunction-type genus constraints.
- Demonstrates cases where traditional invariants vanish but the new invariant does not.

## Abstract

We construct an invariant of closed ${\rm spin}^c$ 4-manifolds using families of Seiberg-Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a 4-manifold. We also give examples of 4-manifolds which admit positive scalar curvature metrics and for which this invariant does not vanish. This non-vanishing result of our invariant provides a new class of adjunction-type genus constraints on configurations of embedded surfaces in a 4-manifold whose Seiberg-Witten invariant vanishes.