A cohomological Seiberg–Witten invariant emerging from the adjunction inequality

摘要

Abstract We construct an invariant of closed spinc$mathrm{spin}^c$ 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.