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

中文翻译：

---

从附加不等式中出现的上同调 Seiberg-Witten 不变量

我们构造一个封闭的不变量${\mathrm{旋转}}^C$${\mathrm{spin}}^c$使用 Seiberg-Witten 方程族的 4 流形。这个不变量被表述为某个抽象单纯复形上的一个上同调类，该复形由一个 4 流形的嵌入表面组成。我们还给出了 4 流形的示例，这些流形允许正标量曲率度量并且该不变量不会消失。我们的不变量的这种非消失结果为 4 流形中嵌入表面的配置提供了一类新的附加类型属约束，其 Seiberg-Witten 不变量消失。