A cohomological Seiberg–Witten invariant emerging from the adjunction inequality

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