On manifolds with infinitely many fillable contact structures

TL;DR

This paper introduces asymptotically finitely generated contact structures and demonstrates that many manifolds, including cotangent bundles, have infinitely many distinct fillable contact structures, expanding the understanding of symplectic topology.

Contribution

It generalizes Symplectic Homology construction and proves the existence of infinitely many fillable contact structures on broad classes of manifolds.

Findings

Many manifolds have infinitely many fillable contact structures.

Symplectic Homology construction is extended and generalized.

Explicit Hamiltonians are provided for handle attaching invariance.

Abstract

We introduce the notion of asymptotically finitely generated contact structures, which states essentially that the Symplectic Homology in a certain degree of any filling of such contact manifolds is uniformly generated by only finitely many Reeb orbits. This property is used to generalize a famous result by Ustilovsky: We show that in a large class of manifolds (including all unit cotangent bundles and all Weinstein fillable contact manifolds with torsion first Chern class) each carries infinitely many exactly fillable contact structures. These are all different from the ones constructed recently by Lazarev. Along the way, the construction of Symplectic Homology is made more general. Moreover, we give a detailed exposition of Cieliebak's Invariance Theorem for subcritical handle attaching, where we provide explicit Hamiltonians for the squeezing on the handle.