On some examples and constructions of contact manifolds

TL;DR

This paper constructs new examples of high-dimensional contact manifolds with specific properties, including tight virtually overtwisted structures and embeddings, and explores their relations with open books and fiber bundles.

Contribution

It introduces a unified framework for constructing and analyzing contact manifolds, extending known procedures and defining new concepts like Bourgeois contact structures on fiber bundles.

Findings

Existence of tight virtually overtwisted contact manifolds in all dimensions.

Every non-rational homology sphere contact 3-manifold embeds in a hypertight 5-manifold.

A new definition of Bourgeois contact structures that includes previous results and supports open book structures.

Abstract

The first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties. Our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all dimensions and the fact that every closed contact 3-manifold, which is not (smoothly) a rational homology sphere, contact--embeds with trivial normal bundle inside a hypertight closed contact 5-manifold. This uses known construction procedures by Bourgeois (on products with tori) and Geiges (on branched covering spaces). We pass from these procedures to definitions; this allows to prove a uniqueness statement in the case of contact branched coverings, and to study the global properties (such as tightness and fillability) of the results of both constructions without relying on any auxiliary choice in the procedures. A second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book, as introduced by Giroux, and of contact fiber bundle, as introduced by Lerman. For instance, we give a definition of Bourgeois contact structures on flat contact fiber bundles which is local, (strictly) includes the results of the Bourgeois construction, and allows to recover an isotopy class of supporting open books on the fibers. This last point relies on a reinterpretation, inspired by an idea by Giroux, of supporting open books in terms of pairs of contact vector fields.