The Gysin exact sequence for S-equivariant symplectic homology

We define S1-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space. We define a parametrized version of the Robbin-Salamon index, which gives the grading for these new versions of symplectic homology. We indicate several applications and ramifications of our constructions.