Torsion and Open Book Decompositions

TL;DR

This paper explores the properties of open book decompositions in contact 3-manifolds, demonstrating that their binding complements lack Giroux torsion and that a certain invariant of the binding is non-zero, revealing new structural insights.

Contribution

It establishes a connection between open book decompositions and Giroux torsion, and proves the non-vanishing of a sutured Heegaard-Floer invariant for bindings.

Findings

Complement of the binding has no Giroux torsion.

The sutured Heegaard-Floer c-bar invariant of the binding is non-zero.

Provides new structural constraints on contact 3-manifolds with open books.

Abstract

We show that if (B,\pi) is an open book decomposition of a contact 3-manifold (Y,\xi), then the complement of the binding B has no Giroux torsion. We also prove the sutured Heegaard-Floer c-bar invariant of the binding of an open book is non-zero.