The Giroux Correspondence in dimension 3

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.

💡 Research Summary

The paper establishes the Giroux correspondence for all contact 3‑manifolds, extending the authors’ earlier work on tight structures to the over‑twisted case by using Heegaard splittings together with a new “bridging” construction. The classical Giroux theorem states that two open‑book decompositions support isotopic contact structures if and only if they are related by a sequence of positive stabilizations and destabilizations. While this result was already known for tight contact manifolds (L‑V, 2023) and for all dimensions via a high‑dimensional convex‑surface theory (HBH), a low‑dimensional proof that works for arbitrary contact 3‑manifolds was missing.

The authors first recall that any open book determines a canonical Heegaard splitting, and that a convex Heegaard splitting (a Heegaard surface that is convex, with tight handlebodies and product disc systems) determines an open book uniquely up to isotopy. In the tight case, one can pass from one convex Heegaard splitting to another by a sequence of moves that preserve the positive stabilization class of the associated open book. However, for general contact structures intermediate Heegaard splittings may acquire over‑twisted handlebodies, breaking the argument.

To overcome this obstacle the paper introduces “bridging”. A bridge adds a collection of contact 1‑handles to a Heegaard splitting in such a way that both resulting handlebodies become tight, regardless of the original contact structure. The authors denote the bridged splitting by (bH) and show that different choices of bridges do not affect the positive stabilization class of the open book obtained after refinement. Thus any convex Heegaard splitting can be replaced by a highly stabilized, always‑tight version without changing the underlying contact isotopy class.

A second crucial ingredient is a classification of bypass decompositions on product cobordisms. Theorem 2.9 (proved in Appendix A) asserts that for a product ( \Sigma \times I ) any two bypass sequences relating isotopic contact structures differ only by positive (de)stabilizations. The proof is carried out entirely in dimension three, using Legendrian graphs, a 1‑parameter Legendrian approximation theorem, and a new perspective on bypass rotation (Lemma 2.10) that identifies it with Legendrian stabilization. This result supplies the missing “move set” needed to connect different convex Heegaard splittings after bridging.

With these tools the authors complete the argument as follows. Given two convex Heegaard splittings (H) and (H’) of the same contact manifold, they bridge both to obtain (bH) and (bH’). Because the bridged handlebodies are tight, the refinement process of L‑V applies, producing open books ((B,\pi)) and ((B’,\pi’)). By Theorem 2.9 the two open books are related by a sequence of positive (de)stabilizations, which exactly matches the equivalence in Giroux’s original statement. Consequently the original contact structures are isotopic, and the Giroux correspondence holds for arbitrary contact 3‑manifolds.

The paper also includes a careful notation system for Heegaard splittings, refinements, and bridges, a Legendrian‑graph approximation theorem (Theorem 2.1) that allows one to model neighborhoods of graphs by Legendrian graphs, and a discussion of bypass attachment via contact handle theory. The authors emphasize that all arguments stay within the realm of low‑dimensional contact topology, making the proof accessible to readers familiar with Heegaard splittings, convex surface theory, and basic contact handle calculus.

In summary, the work provides a self‑contained, low‑dimensional proof of the Giroux correspondence for all contact 3‑manifolds. It bridges the gap between the tight‑only Heegaard‑splitting approach and the all‑dimensions convex‑surface approach, introduces the bridging technique to guarantee tightness of handlebodies, and supplies a full three‑dimensional proof of the bypass‑move classification needed for the argument. This not only clarifies the relationship between Heegaard splittings and open books but also offers tools that are likely to find further applications in 3‑dimensional contact topology.