$\mathbb{Z}$-环面外部与小结手术四流形

中文摘要 AI 辅助

我们研究一种切割-粘贴操作，其中作者辛建筑块$Y_K$和$X_K$中与亏格一纤维结$K$相关的环面邻域被替换为$S^4$中环面的带标记外部，该外部具有无限循环补群和两个零外围斜率。我们还回顾了从$\Sigma_g\times T^2$到$M_K\times S^1$的精确Luttinger手术实现，将结手术/纤维和与Luttinger手术观点保持在同一框架内。对于三叶结块$Y_K$，两次带标记替换给出一个具有相交形式$H$的单连通流形，因此该流形同胚于$S^2\times S^2$。一次外部粘合给出一个光滑同伦$4$-球面。对于秩六构造，我们使用两个$Y_K\setminus\nu\Sigma_2$副本的恒等加倍，而不是定义原始$X_K$的对合粘合。沿存留的边缘环面进行两次带标记替换给出一个具有$e=8$和$\sigma=0$的单连通流形。一个显式几何基具有相交形式$3H$，因此所得流形同胚于$\\#_3(S^2\times S^2)$。Case II小扰动Seiberg-Witten不变量消失。恒等粘合秩六族的Seiberg-Witten不变量也消失；特别地，这些流形是非辛的。

英文摘要

We study a cut-and-paste operation in which torus neighborhoods in the author's symplectic building blocks $Y_K$ and $X_K$, associated to a genus-one fibered knot $K$, are replaced by marked exteriors of tori in $S^4$ with infinite cyclic complement group and two null peripheral slopes. We also recall the exact Luttinger-surgery realization of $M_K\times S^1$ from $Σ_g\times T^2$, keeping the knot-surgery/fiber-sum and Luttinger-surgery viewpoints in the same framework. For the trefoil block $Y_K$, two marked replacements give a simply connected manifold with intersection form $H$, hence a manifold homeomorphic to $S^2\times S^2$. A one-exterior gluing gives a smooth homotopy $4$-sphere. For the rank-six construction we use the identity double of two copies of $Y_K\setminusνΣ_2$, rather than the involutive gluing defining the original $X_K$. Two marked replacements along the surviving rim tori give a simply connected manifold with $e=8$ and $σ=0$. An explicit geometric basis has intersection form $3H$, so the resulting manifold is homeomorphic to $\#_3(S^2\times S^2)$. The Case II small-perturbation Seiberg--Witten invariant vanishes. The Seiberg--Witten invariant of the identity-glued rank-six family also vanishes; in particular, these manifolds are nonsymplectic.

发表机构

• School of Mathematics, University of Minnesota（明尼苏达大学数学学院）
• Department of Mathematics, Harvard University（哈佛大学数学系）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

36 pages, 4 figures