Holomorphic curves in the symplectizations of lens spaces: an elementary approach

TL;DR

This paper develops an elementary method to study rational pseudo-holomorphic curves in symplectizations of lens spaces, leading to a new proof of lens space classification based on contactomorphisms and symplectic techniques.

Contribution

It introduces a computational scheme for moduli spaces of holomorphic curves in lens spaces and uses it to classify lens spaces via contactomorphisms with a purely symplectic approach.

Findings

Established regularity of the standard almost complex structure.

Proved that contactomorphisms imply specific congruences between lens space parameters.

Provided a symplectic proof of the lens space classification theorem.

Abstract

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