Cylindrical contact homology of 3–dimensional Brieskorn 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 $\frac{1}{a_1} + \cdots + \frac{1}{a_n} < n-2$.