Contact structures on principal circle bundles

TL;DR

This paper characterizes when principal circle bundles over even-dimensional manifolds admit invariant contact structures, showing that all such bundles over 4-manifolds do, and connects these results to recent advances in symplectic filling theory.

Contribution

It provides a necessary and sufficient condition for invariant contact structures on principal circle bundles, extending understanding of contact geometry in higher dimensions.

Findings

All circle bundles over 4-manifolds admit invariant contact structures.

Only trivial bundles over a given base carry invariant contact structures.

Connections to recent work on weak symplectic fillings in higher dimensions.

Abstract

We describe a necessary and sufficient condition for a principal circle bundle over an even-dimensional manifold to carry an invariant contact structure. As a corollary it is shown that all circle bundles over a given base manifold carry an invariant contact structure, only provided the trivial bundle does. In particular, all circle bundles over 4-manifolds admit invariant contact structures. We also discuss the Bourgeois construction of contact structures on odd-dimensional tori in this context, and we relate our results to recent work of Massot, Niederkrueger and Wendl on weak symplectic fillings in higher dimensions.