Symplectic homology of some Brieskorn manifolds

TL;DR

This paper uses symplectic homology to distinguish contact structures on Brieskorn manifolds and proves the existence of infinitely many exotic contact structures on S^7, expanding understanding of contact topology.

Contribution

It demonstrates symplectic homology's effectiveness in distinguishing contact structures and constructs infinitely many exotic contact structures on S^7.

Findings

Symplectic homology distinguishes contact structures on certain Brieskorn manifolds.

Existence of infinitely many exotic contact structures on S^7.

Construction of exotic structures using 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 $\Sigma(2l,2,2,2)$, 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 $\Sigma(78k+1,13,6,3,3)$. We end with some considerations about extending this result to higher dimensions.