On the Bauer-Furuta and Seiberg-Witten invariants of families of 4‐manifolds
On the Bauer-Furuta and Seiberg-Witten invariants of families of 4‐manifolds
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关于 4 流形族的 Bauer-Furuta 和 Seiberg-Witten 不变量
DOI：
10.1112/topo.12229
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发表时间：
2022
影响因子：
1.1
通讯作者：
Konno Hokuto
中科院分区：
文献类型：
--
作者：
Baraglia David;Konno Hokuto
我们证明了光滑4‐流形族的Seiberg-Witten族不变量如何通过上同调公式从Bauer-Furuta族不变量中恢复出来。我们用这个公式推导出Seiberg-Witten不变量族的几个性质。我们给出了Seiberg-Witten不变量族的Steenrod平方的一个公式，从而得到了这些不变量与该族的自旋$^c$索引束的Chern类之间的一系列模2关系。结果，我们发现了4 -流形X$X$的普通Seiberg-Witten不变量的一个新方面：它们阻碍了某些纤维与X$X$微分同构的4 -流形族的存在。作为一个具体的几何应用，我们将检测K3$K3$曲面的非光滑族。我们的形式主义也导致了一个简单的新证明的家庭越过墙公式。最后，我们引入K$K$理论Seiberg-Witten不变量，并给出了用上同调Seiberg-Witten不变量表示K$K$理论Seiberg-Witten不变量的陈氏性质的公式。这导致了Seiberg-Witten不变量族的新的可整除性质。
We show how the families Seiberg–Witten invariants of a family of smooth 4‐manifolds can be recovered from the families Bauer–Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg–Witten invariants. We give a formula for the Steenrod squares of the families Seiberg–Witten invariants leading to a series of mod 2 relations between these invariants and the Chern classes of the spinc$^c$ index bundle of the family. As a result, we discover a new aspect of the ordinary Seiberg–Witten invariants of a 4‐manifold X$X$: they obstruct the existence of certain families of 4‐manifolds with fibres diffeomorphic to X$X$. As a concrete geometric application, we shall detect a non‐smoothable family of K3$K3$ surfaces. Our formalism also leads to a simple new proof of the families wall crossing formula. Lastly, we introduce K$K$‐theoretic Seiberg–Witten invariants and give a formula expressing the Chern character of the K$K$‐theoretic Seiberg–Witten invariants in terms of the cohomological Seiberg–Witten invariants. This leads to new divisibility properties of the families Seiberg–Witten invariants.