5-dimensional Bourgeois contact structures are tight

TL;DR

This paper proves that all 5-dimensional Bourgeois contact structures, constructed from a supporting open book decomposition, are universally tight regardless of the tightness or overtwistedness of the original contact manifold.

Contribution

It establishes the universal tightness of all 5-dimensional Bourgeois contact structures, a result previously unknown in contact topology.

Findings

All such structures are universally tight in dimension 5.

The tightness holds regardless of the original manifold's tightness.

The result applies to both tight and overtwisted initial contact manifolds.

Abstract

Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction for a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is tight or overtwisted.