A cohomological Seiberg-Witten invariant emerging from the adjunction inequality

Abstract

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

arXiv comment55 pages, 8 figures, Subsection 2.2 was added, accepted for publication by the Journal of Topology