Diagrams for contact 5-manifolds

TL;DR

This paper introduces handle move techniques for Kirby diagrams of Stein surfaces in 5-dimensional contact manifolds, leading to classification results and new insights into contact structures on 5-manifolds.

Contribution

It develops handle move methods on Kirby diagrams for Stein surfaces in contact 5-manifolds and applies them to classify subcritically Stein fillable structures and characterize the 5-sphere.

Findings

Classification of subcritically Stein fillable contact 5-manifolds

Characterization of the standard contact structure on the 5-sphere

New existence proof for contact structures on simply connected 5-manifolds

Abstract

According to Giroux, contact manifolds can be described as open books whose pages are Stein manifolds. For 5-dimensional contact manifolds the pages are Stein surfaces, which permit a description via Kirby diagrams. We introduce handle moves on such diagrams that do not change the corresponding contact manifold. As an application, we derive classification results for subcritically Stein fillable contact 5-manifolds and characterise the standard contact structure on the 5-sphere in terms of such fillings. This characterisation is discussed in the context of the Andrews-Curtis conjecture concerning presentations of the trivial group. We further illustrate the use of such diagrams by a covering theorem for simply connected spin 5-manifolds and a new existence proof for contact structures on simply connected 5-manifolds.