# Bourgeois contact structures: tightness, fillability and applications

## TL;DR

This paper studies Bourgeois contact structures, proving their universal tightness in dimension 5, providing obstructions to symplectic fillability in higher dimensions, and classifying certain contact manifolds.

## Contribution

It establishes universal tightness of Bourgeois structures in dimension 5 and introduces obstructions to strong fillability in higher dimensions, expanding understanding of contact manifold fillability.

## Key findings

- All Bourgeois structures in dimension 5 are universally tight.
- Identifies obstructions to strong symplectic fillings in higher dimensions.
- Classifies the symplectic fillability of the unit cotangent bundle of the n-torus.

## Abstract

Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction of 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 itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact $5$-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for $\mathbb S^1$-invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the $n$-torus has a unique symplectically aspherical strong filling up to diffeomorphism.