Symplectic homology of some Brieskorn manifolds

This paper consists of two parts. In the first part, we use symplectic homology to distinguish the contact structures on the Brieskorn manifolds $\mathrm{\Sigma }(2l,2,2,2)$, which contact homology cannot distinguish. This answers a question from [22]. In the second part, we prove the existence of infinitely many exotic but homotopically trivial exotic contact structures on $S^7$, distinguished by the mean Euler characteristic of $S^1$-equivariant symplectic homology. Apart from various connected sum constructions, these contact structures can be taken from the Brieskorn manifolds $\mathrm{\Sigma }(78k+1,13,6,3,3)$. We end with some considerations about extending this result to higher dimensions.