The Giroux Correspondence in dimension 3

Abstract

This paper proves the Giroux Correspondence in dimension three using Heegaard splittings of contact manifolds. In two of the authors earlier paper they proved the Giroux Correspondence for tight contact 3-manifolds via convex Heegaard surfaces, and simultaneously, Honda, Breen and Huang gave an alldimensions proof of the Giroux Correspondence by generalising convex surface theory to higher dimensions. This paper extends the Heegaard splitting approach to arbitrary (not necessarily tight) contact 3-manifolds in order to provide a proof accessible to a low-dimensional audience. The proof assumes classification moves relating bypass decompositions for isotopic contact structures on cobordisms that are topological products; in the Appendix, we prove this result in the 3- dimensional setting.

Figures (26)

• Figure 1: Left: a bypass half-disc attached along the arc $c$. Centre: the attached $1$-handle is a regular neighbourhood of an arc Legendrian isotopic to $c$. Right: attaching the shaded $2$-handle along $c\cup c'$ completes the bypass attachment.
• Figure 2: Left: Local model for a bypass attachment arc. Centre: The new dividing set after attaching a bypass along the arc shown on the left. Right: A trivial bypass arc. Attaching a bypass along the arc shown preserves the dividing set up to isotopy.
• Figure 3: Left: The bypass arc $c_T$ is a rotation of the bypass arc $c$. Left centre: After attaching a bypass along $c$, $c_T$ becomes a trivial bypass arc. Right centre: The dividing set is preserved up to isotopy by attaching a bypass to $c_T$. Right: The bypass disc attached along $c_T$ can be constructed in a neighbourhood of $R'$ and the bypass disc for $c$.
• Figure 4: Attach a $1$-handle $h_T^1$ along a push-off of the Legendrian curve $d$. Then isotope the disc cobounded by $h_T^1$ and $d$ across the product disc $A$ to produce a bypass disc $D$ cobounded by $h_T^1$ and $c_T$.
• Figure 5: To refine a tight Heegaard splitting $\mathcal{H}$, stabilise it via tunnels drilled through non-product discs to produce a convex splitting $\widetilde{H}$. Here, the meriodonal disc $A$ in $\mathcal{H}$ with $|\partial A \cap \Gamma_\Sigma|=4$ splits into a pair of product discs $A_1$, $A_2$ in $\widetilde{\mathcal{H}}$.

Theorems & Definitions (66)

• Theorem 1.1
• Theorem 2.1
• Corollary 2.2
• proof : Proof of Theorem \ref{['thm:Legapprox1']}
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Definition 2.6
• Proposition 2.7
• Theorem 2.8