The Gysin exact sequence for $S^1$-equivariant symplectic homology

We define $S^1$-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.
Comment: v3: 48 pages. Added Lemma 5.8 relating the long exact sequence of the cone and the Gysin sequence. Various other modifications throughout the text following referee's comments. This is the final version published in J. Topol. Anal