发表机构
University of Minnesota; Harvard University（明尼苏达大学; 哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明纤维化椭圆对合给出零手术结与环面乘积的二重分支覆盖，计算编织词，并推广至亏格二纤维化结，同时给出扩展障碍。
AI 中文摘要
设 $K$ 为三叶结或八字结，$M_K$ 表示 $K$ 上的零手术。我们证明纤维化椭圆对合给出一个二重分支覆盖 $M_K\times S^1\longrightarrow T^2\times S^2$，其分支轨迹是一个截面以及一个连通的编织三截面。对两个结显式计算出相应的编织词，并使之与沿乘积拉格朗日环面的 Luttinger 手术的关系可见。更一般地，对于具有超椭圆单值化的亏格-$g$ 纤维化结，同样的构造给出一个编织的 $(2g+2)$-截面；这包括每个亏格二纤维化结。我们还陈述了相应的 $T^2$ 上的曲面丛版本。然后，我们在 $Y_K$ 及其扭曲双 $X_K$ 的构造中追踪诱导的外围数据和粘合微分同胚，这些构造由作者在 \cite[Section~4]{Akh07} 中研究，并通过与 $\CP^2\\#k\CPb^2$（$k=3,5$）同胚的小 exotic 流形的构造 \cite{AkhmedovSmall,ABP,AP2008} 进行。在 $k=5$ 例子的对称构造 \cite[Section~3.3]{ABP} 中，每个被加数允许一个沿曲面分支的二重覆盖，尽管在扭曲双中使用的标记补空间不需要不变。最后，我们证明了边界作用、相对同调和局部光滑化障碍，这些障碍阻止将这些局部对合扩展穿过粘合区域到闭流形的分支覆盖。
英文摘要
Let $K$ be the trefoil or figure-eight knot and let $M_K$ denote zero surgery on $K$. We prove that the fiberwise elliptic involution gives a two-fold branched cover \[ M_K\times S^1\longrightarrow T^2\times S^2 \] whose branch locus is a section together with a connected braided three-section. The corresponding braid words are computed explicitly for both knots and make visible the relation with Luttinger surgery along product Lagrangian tori. More generally, for a genus-$g$ fibered knot with hyperelliptic monodromy, the same construction gives a braided $(2g+2)$-section; this includes every genus-two fibered knot. We also state the corresponding surface-bundle version over $T^2$. We then track the induced peripheral data and gluing diffeomorphisms in the constructions of $Y_K$ and its twisted double $X_K$, as studied by the author in \cite[Section~4]{Akh07}, and through the constructions of the small exotic manifolds homeomorphic to $\CP^2\#k\CPb^2$, $k=3,5$ \cite{AkhmedovSmall,ABP,AP2008}. In the symmetric construction of a $k=5$ example \cite[Section~3.3]{ABP}, each summand admits a double cover branched along a surface, although the marked complements used in the twisted double need not be invariant. Finally, we prove boundary-action, relative-homology, and local-smoothing obstructions to extending these local involutions across the gluing regions to a branched cover of the closed manifold.
Comments26 pages, 3 figures
