E8-plumbings and exotic contact structures on spheres

Abstract

We prove the existence of exotic but homotopically trivial contact structures on spheres of dimension 8k−1. Together with previous results of Eliashberg and the second author, this proves the existence of such structures on all odd-dimensional spheres. Here, a contact structure is called “exotic” if it is not diffeomorphic to the standard contact structure; “homotopically trivial” means that it induces, like the standard structure, an almost complex structure along the sphere that extends over a cobounding disc. The construction is based on finding certain highly connected symplectic manifolds, with boundary a sphere, via plumbings of disc bundles. Exoticity of the induced contact structure on the boundary follows from a result of Eliashberg, Floer, Gromov, and McDuff about symplectic fillings of the standard contact structure.