Contact domination

TL;DR

The paper proves that every closed connected oriented odd-dimensional manifold can be dominated by a tight contact manifold, extending symplectic domination results to contact geometry and discussing fillability and applications.

Contribution

It establishes the existence of contact domination for all odd-dimensional manifolds and explores fillability properties and applications to contact divisors.

Findings

Every odd-dimensional manifold admits a contact domination.

The dominating contact manifold can be Liouville-fillable but not Weinstein-fillable.

Application to contact divisors as zero sets of asymptotically contact-holomorphic sections.

Abstract

In this note, we prove that every closed connected oriented odd-dimensional manifold admits a map of non-zero degree (i.e., a domination) from a tight contact manifold of the same dimension. This provides an odd-dimensional counterpart of a symplectic domination result due to Joel Fine and Dmitri Panov. We prove that the dominating contact manifold can be ensured to be Liouville-fillable, but not Weinstein-fillable in general. We discuss an application for contact divisors arising as zero sets of asymptotically contact-holomorphic sections.