[Paper Review] On the contact class in Heegaard Floer homology

Ko Honda, William Kazez+1 more|ArXiv.org|Sep 26, 2006

Geometric and Algebraic Topology11 references17 citations

TL;DR

This paper presents an alternative, geometrically intuitive description of the Ozsváth-Szabó contact class in Heegaard Floer homology using the EH class, which is shown to coincide with the original contact invariant. The key result establishes that for a once-punctured torus open book, the contact structure is tight if and only if the monodromy is right-veering, providing a complete characterization of tightness in this case.

ABSTRACT

We present an alternate description of the Ozsvath-Szabo contact class in Heegaard Floer homology. Using our contact class, we prove that if a contact structure (M,ξ) has an adapted open book decomposition whose page S is a once-punctured torus, then the monodromy is right-veering if and only if the contact structure is tight.

Motivation & Objective

• To provide a more geometric, hands-on description of the Ozsváth-Szabó contact class in Heegaard Floer homology.
• To establish a link between right-veering monodromies and tight contact structures on 3-manifolds.
• To prove a complete characterization of tightness in terms of monodromy behavior for open books with once-punctured torus pages.
• To resolve a conjecture that minimal right-veering open books support tight contact structures in the genus one case.

Proposed method

• Introduce the EH class in Heegaard Floer homology as an alternative to the Ozsváth-Szabó contact class.
• Prove that the EH class equals the original contact class $ c(\xi) $, establishing it as a contact invariant.
• Use a specific Heegaard diagram for once-punctured torus open books with pseudo-Anosov monodromy.
• Apply holomorphic disk counting arguments in symmetric products of surfaces to show non-vanishing of the EH class.
• Analyze the dynamics of monodromy via Dehn twists and fractional Dehn twist coefficients in the mapping class group.
• Use Farey tessellation and conjugacy in $ SL(2,\mathbb{Z}) $ to choose convenient bases for computing the contact invariant.

Experimental results

Research questions

• RQ1 Can the Ozsváth-Szabó contact class in Heegaard Floer homology be re-expressed in a more geometric, computable form?
• RQ2 Is there a direct relationship between right-veering monodromies and tight contact structures on 3-manifolds?
• RQ3 For open books with once-punctured torus pages, does right-veering monodromy fully characterize tight contact structures?
• RQ4 Does the EH class provide a computable obstruction to overtwistedness in this setting?
• RQ5 What is the precise role of the fractional Dehn twist coefficient in determining the non-vanishing of the contact class?

Key findings

• The EH class in $ \widehat{HF}(-M) $ is proven to equal the Ozsváth-Szabó contact class $ c(\xi) $, confirming it as a contact invariant.
• For a once-punctured torus open book, the contact structure is tight if and only if the monodromy is right-veering.
• When the monodromy is pseudo-Anosov with fractional Dehn twist coefficient $ c = \frac{1}{2} $, the contact class is non-zero, implying tightness.
• For reducible monodromies, the contact class is non-zero if the monodromy is right-veering, even when negative Dehn twists are present.
• The EH class vanishes for non-right-veering monodromies, confirming overtwistedness in such cases.
• The proof relies on showing the absence of holomorphic disks in the Heegaard diagram that would otherwise kill the generator $ \mathbf{x} = (x_0, y_0) $, ensuring non-triviality of the class.