On properties of Bourgeois contact structures

TL;DR

This paper investigates how certain properties of contact manifolds are preserved or not when applying the Bourgeois construction to produce new contact structures on the product of a manifold with a torus, using Giroux's open book formalism.

Contribution

It analyzes the inheritance of properties under the Bourgeois construction and clarifies the use of Giroux's contact open book framework through examples.

Findings

Some properties of the original contact manifold are preserved in the product.

Certain properties are not inherited by the Bourgeois contact structure.

The paper provides illustrative examples using Giroux's formalism.

Abstract

The Bourgeois construction associates to every contact open book on a manifold $V$ a contact structure on $V\times T^2$. We study in this article some of the properties of $V$ that are inherited by $V\times T^2$ and some that are not. Giroux has provided recently a suitable framework to work with contact open books. In the appendix of this article, we quickly review this formalism, and we work out a few classical examples of contact open books to illustrate how to use this new language.