Spectral order for contact manifolds with convex boundary

TL;DR

This paper extends the spectral order concept for contact 3-manifolds with convex boundary, showing how it bounds overtwisted and Giroux torsion structures, linking spectral order to contact topology invariants.

Contribution

It generalizes the spectral order to manifolds with boundary and establishes bounds related to overtwisted disks and Giroux torsion domains.

Findings

Overtwisted contact structures have spectral order zero.

The spectral order of a perturbation of a Giroux torsion domain is at most two.

Contact structures with positive Giroux torsion have spectral order at most two.

Abstract

We extend the Heegaard Floer homological definition of spectral order for closed contact 3-manifolds due to Kutluhan, Mati\'c, Van Horn-Morris, and Wand to contact 3-manifolds with convex boundary. We show that the order of a codimension zero contact submanifold bounds the order of the ambient manifold from above. As the neighborhood of an overtwisted disk has order zero, we obtain that overtwisted contact structures have order zero. We also prove that the order of a small perturbation of a $2\pi$ Giroux torsion domain has order at most two, hence any contact structure with positive Giroux torsion has order at most two (and, in particular, a vanishing contact invariant).