Symplectic homology and 3-dimensional Besse manifolds with vanishing first Chern class

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.