Bourgeois' contact manifolds are tight

TL;DR

This paper proves that Bourgeois' contact structures on certain manifolds are always tight, using contact homology and holomorphic foliation techniques.

Contribution

It establishes the tightness of Bourgeois' contact structures on $M imes ext{T}^2$, a result not previously known, via novel contact homology computations.

Findings

Bourgeois' contact structures are tight on $M imes ext{T}^2$.

The proof employs holomorphic foliations and Kuranishi structures.

Contact homology computations confirm tightness.

Abstract

We prove that Bourgeois' contact structures on $M \times \mathbb{T}^{2}$ determined by the supporting open books of a contact manifold $(M, \xi)$ are always tight. The proof is based on a contact homology computation leveraging holomorphic foliations and Kuranishi structures.