On binding sums of contact manifolds

TL;DR

This paper explores how binding sums of contact 3-manifolds can alter properties like tightness and fillability, proving vanishing of contact invariants in certain cases and correcting previous computational errors.

Contribution

It provides examples showing property loss under binding sums, proves vanishing of contact invariants for specific sums, and rectifies a spectral order computation error.

Findings

Binding sums can destroy tightness and fillability.

Vanishing of Heegaard Floer contact invariant in certain sums.

Correction of a spectral order calculation error.

Abstract

In this short note, we give examples of binding sums of contact 3-manifolds that do not preserve properties such as tightness or symplectic fillability. We also prove vanishing of the Heegaard Floer contact invariant for an infinite family of binding sums where the summands are Stein fillable. This recovers a result of Wendl and Latschev-Wendl. Along the way, we rectify a subtle computational error in a paper of Juhasz-Kang concerning the spectral order of a neighbourhood of a Giroux torsion domain.