Nonexistence and existence of fillable contact structures on 3-manifolds

TL;DR

This paper constructs hyperbolic 3-manifolds without fillable contact structures via Dehn surgeries, and shows that large surgeries on certain knots admit Stein fillable structures, providing evidence for the high surgery conjecture.

Contribution

It demonstrates the existence and nonexistence of fillable contact structures on 3-manifolds through explicit constructions and surgeries, advancing understanding of contact topology.

Findings

Many hyperbolic 3-manifolds admit no fillable contact structures.

Large Dehn surgeries on certain knots admit Stein fillable contact structures.

Supports the high surgery conjecture by showing fillability depends on surgery coefficients.

Abstract

In the first part of this paper, we construct infinitely many hyperbolic closed 3-manifolds which admit no symplectic fillable contact structure. All these 3-manifolds are obtained by Dehn surgeries along L-space knots or L-space two-component links. In the second part of this paper, we show that Dehn surgeries along certain knots and links, including those considered in the first part, admit Stein fillable contact structures as long as the surgery coefficients are sufficiently large. This provides some new evidence for the high surgery conjecture raised by Stipsicz.