论文标题
家族塞伯格（Seiberg）的家庭不变性和非透明度循环的差异性循环
The Family Seiberg-Witten Invariant and nonsymplectic loops of diffeomorphisms
论文作者
论文摘要
通过将Kronheimer-Mrowka的结果扩展到家庭环境中，我们证明了Seiberg-inbent的不变性的胶合公式。该公式使一个公式可以通过将其沿3个manifolds的产品家族切开，并研究单极浮子（CO）同源性的诱导地图，从而计算出平滑的4个manifolds家族的不变性家族。当切割3个manifold是L空间时，该公式意味着塞伯格（Seiberg）的家庭不变式，纤维的塞伯格（Seiberg）和家庭dirac操作员索引之间的关系。我们使用这种关系来计算Seiberg-witten的4个manifold家族的不变性，这些家族在使用Hyperkähller家族的奇异结构家族来解决ADE奇异性时会出现。获得了几种应用程序。首先，我们建立了一个简单连接的4个manifolds $ m $（例如所有椭圆表面）的大家庭，以便$π_{1}（\ textrm {diff}（m））$具有$ \ mathbb {Z}^{\ mathbb {z}^{\ infty} $ - summand。对于此类$ M $，产品$ s^{2} \ times m $平稳地纤维在$ s^{2} $上，带有纤维$ m $以无限的不同方式。其次，我们表明，在任何封闭的符号4个manifold上，它包含一个平滑的自我交流$ -1 $ -1 $或$ -2 $的球体，存在一个差异性循环，与符号术的循环无关。这概括了Smirnov的先前结果，并确认了McDuff在维度4中的猜想。它还提供了许多4个manifolds的新示例，其符号形式的空间具有非平凡的基本组或第一同源群。
By extending a result of Kronheimer-Mrowka to the family setting, we prove a gluing formula for the family Seiberg-Witten invariant. This formula allows one to compute the invariant for a smooth family of 4-manifolds by cutting it open along a product family of 3-manifolds and studying the induced maps on monopole Floer (co)homology. When the cutting 3-manifold is an L-space, this formula implies a relation between the family Seiberg-Witten invariant, the Seiberg-Witten invariant of the fiber and the index of the family Dirac operator. We use this relation to calculate the Seiberg-Witten invariant of families of 4-manifolds that arise when resolving an ADE singularity using a hyperkähler family of complex structures near the singularity. Several applications are obtained. First, we establish a large family of simply-connected 4-manifolds $M$ (e.g. all elliptic surfaces) such that $π_{1}(\textrm{Diff}(M))$ has a $\mathbb{Z}^{\infty}$-summand . For such $M$, the product $S^{2}\times M$ smoothly fibers over $S^{2}$ with fiber $M$ in infinitely many distinct ways. Second, we show that on any closed symplectic 4-manifold that contains a smoothly embedded sphere of self-intersection $-1$ or $-2$, there is a loop of diffeomorphisms that is not homotopic to a loop of symplectormorphisms. This generalizes a previous result by Smirnov and confirms a conjecture by McDuff in dimension 4. It also provides many new examples of 4-manifolds whose space of symplectic forms has a nontrivial fundamental group or first homology group.
