Contact surgery, open books, and symplectic cobordisms

In this thesis, we study contact manifolds and symplectic cobordisms between them using open book decompositions and various types of symplectic handle attachment. In the first half, we describe two algorithm which interpolate between open book decompositions and contact surgery diagrams of a contact 3-manifold. These algorithms are used to (1) establish the fact -- originally due to Ding-Geiges -- that every contact 3-manifold admits a contact surgery presentation, (2) bound (and sometimes compute) the support invariants -- due to Etnyre-Ozbagci -- of certain contact 3-manifolds and Legendrian links, and (3) describe a "Kirby calculus" for Legendrian links in the standard contact 3-sphere. In the second half of the thesis, we describe a new surgery operation for contact manifolds (of arbitrary dimension) called the \emph{Liouville connect sum}, which generalizes Weinstein handle attachment. Using this surgery operation we (1) generalize results of Baker-Etnyre-van Horn-Morris and Baldwin concerning the existence of "monodromy multiplication cbordisms", (2) study the fillability of certain contact-branched covers, and (3) generalize the definition of contact (1/k)-surgery to arbitrary dimensions. Finally, we use open books and symplectic fillings to show that while the square of a Dehn twist about an n-sphere is smoothly isotopic to the identity in if and only if n is 2 or 6, no power of a Dehn twist is ever symplectically isotopic to the identity mapping. This generalizes results of Seidel to all dimensions.

本论文借助开书分解（open book decompositions）与各类辛把手附件（symplectic handle attachment），研究接触流形（contact manifolds）及其间的辛配边（symplectic cobordisms）。论文前半部分阐述了两种可实现接触三维流形（contact 3-manifold）的开书分解与接触手术图（contact surgery diagrams）之间插值转换的算法。这些算法可用于：(1) 证实由Ding-Geiges首次提出的论断：任意接触三维流形均可接受接触手术表示；(2) 对特定接触三维流形与Legendrian纽结（Legendrian links）的埃特内尔-奥兹巴奇（Etnyre-Ozbagci）支撑不变量（support invariants）进行估值（部分情形下可计算出具体数值）；(3) 针对标准接触3-球面（standard contact 3-sphere）中的Legendrian纽结，构建一套"基尔比演算（Kirby calculus）"方法。论文后半部分针对任意维数的接触流形，提出一种名为**刘维尔连通和（Liouville connect sum）**的新型手术操作，该操作是温斯坦把手附件（Weinstein handle attachment）的推广。借助该手术操作，我们：(1) 推广了Baker-Etnyre-van Horn-Morris与Baldwin关于"单模乘法配边（monodromy multiplication cobordisms）"存在性的相关结论；(2) 研究了特定接触分支覆盖（contact-branched covers）的可填充性；(3) 将接触(1/k)-手术（contact (1/k)-surgery）的定义推广至任意维数。最后，我们借助开书分解与辛填充（symplectic fillings）证明：尽管n维球面的德恩扭转（Dehn twist）的平方当且仅当n为2或6时光滑同痕（smoothly isotopic）于恒等映射（identity mapping），但不存在任何次幂的德恩扭转能辛同痕（symplectically isotopic）于恒等映射。这一结论将Seidel的相关研究成果推广至所有维数。