Symplectic homology of some Brieskorn manifolds

Abstract

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 , 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 Σ(78k + 1, 13, 6, 3, 3). We end with some considerations about extending this result to higher dimensions.

Rabinowitz–Floer Homology on Brieskorn Spheres: Fig. 1.

“…The original idea of the following proof is due to Peter Uebele, who showed a similar result for symplectic homology in [50].…”

“…Together with Peter Uebele, [50], we define in the σ-symmetric case when im(v) ⊂ V f ix the set of symmetric regular points by…”

“…In the following we will determine the 1-periodic orbits of X ψ and calculate their Conley-Zehnder indices. By (50), the Hamiltonian vector field X ψ is given by…”

Vanishing of Symplectic Homology and Obstruction to Flexible Fillability

“…, 2)) = 0. Such argument can be applied to many Brieskorn manifolds, whose (positive) symplectic cohomology have been computed [12,21]. The argument might be possible to generalize to all Brieskorn manifolds by studying the Conley-Zehnder indexes carefully.…”

Reducibility in Sasakian geometry

“…The fact that the contact structures L(2, 2, 2, 2k) are inequivalent for different k has recently been proven by Uebele[Ueb16] using the plus part of non-equivariant symplectic homology on a convenient filling which Uebele shows is a contact invariant in this case. Interestingly these contact structures cannot be distinguished by their mean Euler characteristic nor the plus part of the S 1 -equivariant symplectic homology which are known contact invariants[KvK16,BMvK16].…”