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 Σ ( 2 ℓ , 2 , 2 , 2 ) , which contact homology cannot distinguish. This answers a question from Kwon and van Koert (Brieskorn manifolds in contact topology, preprint, 2013 . arXiv:1310.0343 ). 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 Σ ( 78 k + 1 , 13 , 6 , 3 , 3 ) . We end with some considerations about extending this result to higher dimensions.