论文标题
镜头空间符号的全态曲线：一种基本方法
Holomorphic curves in the symplectizations of lens spaces: an elementary approach
论文作者
论文摘要
我们提出了一个基本计算方案，用于在3维透镜空间的符号化中的有理伪旋晶曲线的模量空间，该曲线配备了由$ S^3 $上的标准Morse-Bott接触形式引起的Morse-Bott接触表格。作为一个应用程序，我们证明，如果$ p $ prime和$ 1 <q，q'<p-1 $，如果镜头空间$ l（p，q）$和$ l（p，q'）$之间存在接触型，其中两个空间都配备了其标准接触结构，则$ q \ equiv（q'equiv（q'q''）^{\ equ'）为了证明，我们研究了两条裤子的模量空间，并详细介绍了两个非摘要的末端，并确定使用的标准几乎复杂的结构是常规的。然后，接触型的存在使我们能够遵循颈部拉伸过程，通过该过程，我们比较了在$ L（p，q）$和$ l（p，q，q'）$的符合裤子中编码的同件关系。将我们的证明与本田关于透镜空间上普遍紧密接触结构的分类的结果相结合，我们提供了上面提到的类中镜头空间的透明镜头分类的纯粹合成/接触拓扑证明
We present an elementary computational scheme for the moduli spaces of rational pseudo-holomorphic curves in the symplectizations of 3-dimensional lens spaces, which are equipped with Morse-Bott contact forms induced by the standard Morse-Bott contact form on $S^3$. As an application, we prove that for $p$ prime and $1<q,q'<p-1$, if there is a contactomorphism between lens spaces $L(p,q)$ and $L(p,q')$, where both spaces are equipped with their standard contact structures, then $q\equiv (q')^{\pm 1}$ in$\mod p$. For the proof we study the moduli spaces of pair of pants with two non-contractible ends in detail and establish that the standard almost complex structure that is used is regular. Then the existence of a contactomorphism enables us to follow a neck-stretching process, by means of which we compare the homotopy relations encoded at the non-contractible ends of the pair of pants in the symplectizations of $L(p,q)$ and $L(p,q')$. Combining our proof with the result of Honda on the classification of universally tight contact structures on lens spaces, we provide a purely symplectic/contact topological proof of the diffeomorphism classification of lens spaces in the class mentioned above
