Cylindrical contact homology of 3-dimensional Brieskorn manifolds

TL;DR

This paper computes the cylindrical contact homology for a class of 3-dimensional Brieskorn manifolds with natural contact structures, extending the understanding of symplectic invariants in these geometric settings.

Contribution

It provides a complete description of cylindrical contact homology for Brieskorn 3-manifolds under specific conditions, advancing the computation of symplectic invariants.

Findings

Explicit formulas for cylindrical contact homology of Brieskorn manifolds.

Verification of invariance conditions for the contact homology.

Extension of known results to a broader class of 3-manifolds.

Abstract

Cylindrical contact homology is a comparatively simple incarnation of symplectic field theory whose existence and invariance under suitable hypotheses was recently established by Hutchings and Nelson. We study this invariant for a general Brieskorn 3-manifold $\Sigma(a_1,\ldots, a_n)$, and give a complete description of the cylindrical contact homology for this 3-manifold equipped with its natural contact structure, for any $a_j$ satisfying $\frac{1}{a_1} + \cdots + \frac{1}{a_n} < n-2$.