Symplectic Homology and 3-dimensional Besse Manifolds with vanishing first Chern class

TL;DR

This paper explores symplectic homology as an invariant for 3D Besse manifolds with trivial first Chern class, classifying such manifolds and computing related indices and invariants.

Contribution

It introduces a method to classify Besse manifolds with vanishing first Chern class and computes symplectic homology and indices for these structures.

Findings

Classification of Besse manifolds with zero first Chern class

Computation of Robbin-Salamon indices for periodic Reeb orbits

Calculation of symplectic homology for specific cases

Abstract

In this paper, we will show that certain types of symplectic homology can be used as an invariant of 3-dimensional Besse manifolds, which are strict contact manifolds with periodic Reeb flow. For simplicity, we will assume our Besse structures to be a trivial plane bundle. To identify Besse manifolds with such a condition, we actually compute the first Chern class of each Besse structure and classify the Besse manifolds with vanishing first Chern class. We will also compute Robbin-Salamon indices of periodic Reeb orbits in Besse manifolds, and symplectic homology (of its filling) of certain cases. From its definition, Besse manifolds naturally become Seifert fibration and thus one can extract invariants such as the orbifold Euler characteristic and the Euler number of this Seifert fibration. These invariants will become important in our computation.