[Paper Review] Round surgery and contact structures on 3-manifolds

Jiro Adachi|arXiv (Cornell University)|Mar 12, 2017

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper introduces contact round surgery—a new method to construct and classify contact structures on 3-manifolds by modifying contact structures through round handles of index 1 and 2. It provides an alternative proof of the existence of contact structures on all closed orientable 3-manifolds and shows that every such contact structure arises from the standard contact 3-sphere via a sequence of contact round surgeries, with the Lutz twist and Giroux torsion fully realized within this framework.

ABSTRACT

Contact round surgery of contact 3-manifolds is introduced in this paper. By using this method, an alternative proof of the existence of a contact structure on any closed orientable 3-manifold is given. It is also proved that any contact structure on any closed orientable 3-manifold is constructed from the standard contact structure on the 3-dimensional sphere by contact round surgeries. For the proof, important operations in contact topology, the Lutz twist and the Giroux torsion, are described in terms of contact round surgeries.

Motivation & Objective

• To develop a new surgery-theoretic framework for constructing contact structures on 3-manifolds using round handles.
• To provide an alternative proof of the existence of contact structures on any closed orientable 3-manifold.
• To show that all contact structures on closed orientable 3-manifolds are obtained from the standard contact 3-sphere via contact round surgeries.
• To express fundamental operations in contact topology—Lutz twist and Giroux torsion—within the contact round surgery formalism.

Proposed method

• Contact round surgery is defined by attaching round handles of index 1 and 2 to a contact cobordism, modifying the contact structure along tubular neighborhoods of transverse knots and convex tori.
• The reversibility of contact round surgeries is established by showing that the inverse operation recovers the original contact structure through inverse gluing of removed solid tori and thickened tori.
• The Lutz twist is realized as a sequence of contact round surgeries, enabling the construction of overtwisted contact structures from the standard one.
• Giroux torsion is shown to be realizable via contact round surgeries, linking it to the surgery framework.
• The classification of contact structures on S³ by Eliashberg is applied, using the fact that all such structures are reachable via Lutz twists, which are now represented as surgery sequences.
• The proof leverages Asimov’s result that any closed 3-manifold is obtained from S³ via topological round surgeries, now extended to the contact category.

Experimental results

Research questions

• RQ1 Can contact round surgery be used to construct a contact structure on any closed orientable 3-manifold?
• RQ2 Is every contact structure on a closed orientable 3-manifold obtainable from the standard contact 3-sphere via contact round surgeries?
• RQ3 How can the Lutz twist—a key operation in contact topology—be expressed in terms of contact round surgeries?
• RQ4 What is the relationship between Giroux torsion and contact round surgery?
• RQ5 Can the classification of contact structures on S³ be recovered using contact round surgery operations?

Key findings

• All closed orientable 3-manifolds admit a contact structure, and this result is reproven via contact round surgery.
• Every contact structure on any closed orientable 3-manifold is constructed from the standard contact 3-sphere through a finite sequence of contact round surgeries of index 1 and 2.
• The Lutz twist is realized as a sequence of contact round surgeries, providing a surgery-theoretic interpretation of this fundamental operation.
• Giroux torsion is shown to be realizable within the contact round surgery framework, linking it to the surgery formalism.
• Contact round surgeries are reversible, ensuring that the surgery process is a valid and invertible construction method in the contact category.
• The classification of contact structures on S³—up to isotopy—can be recovered entirely through contact round surgeries, confirming the completeness of the surgery model.