[Paper Review] A unique decomposition theorem for tight contact 3-manifolds

Fan Ding, Hansjörg Geiges|ArXiv.org|Oct 9, 2006

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper establishes a unique decomposition theorem for tight contact 3-manifolds, proving that every non-trivial tight contact 3-manifold admits a unique connected sum decomposition into prime tight contact summands, up to order and contactomorphism. Building on Colin's connected sum construction for tight contact structures, the authors adapt Milnor's uniqueness argument using isotopy invariance and Eliashberg's classification of tight contact structures on the 3-ball to show that the prime summands are uniquely determined by the contact topology.

ABSTRACT

It has been shown by V. Colin that every tight contact 3-manifold can be written as a connected sum of prime manifolds. Here we prove that the summands in this decomposition are unique up to order and contactomorphism.

Motivation & Objective

• To establish a unique decomposition theorem for tight contact 3-manifolds analogous to the prime decomposition of smooth 3-manifolds.
• To resolve the non-uniqueness of the underlying sphere systems in topological prime decompositions by incorporating contact isotopy invariance.
• To prove that the prime tight contact summands in a connected sum decomposition are uniquely determined up to contactomorphism and order.
• To extend Colin’s connected sum construction for tight contact structures to a full uniqueness result by leveraging Eliashberg’s classification of tight contact structures on the 3-ball.

Proposed method

• Adapts Milnor’s uniqueness argument for smooth 3-manifold prime decompositions to the contact setting, using isotopy invariance of contact structures.
• Applies Colin’s connected sum construction for tight contact 3-manifolds, which defines a well-defined contact structure on the connected sum via gluing along 2-spheres.
• Uses Lemma 1 to construct a contact structure on $ S^2 imes [0,1] $ that matches prescribed characteristic foliations on the boundary spheres.
• Employs Eliashberg’s theorem (Theorem 5) to ensure that tight contact structures on the 3-ball are uniquely determined by their boundary characteristic foliation.
• Applies Lemma 2 and Lemma 3 to show that tightness is preserved under isotopy of spheres and that the resulting contact structure is independent of the isotopy class of the embeddings.
• Uses a minimality argument on sphere systems and sphere modifications via isotopy to eliminate redundant spheres, proving uniqueness of the decomposition.

Experimental results

Research questions

• RQ1 Can the connected sum decomposition of a tight contact 3-manifold be made unique up to contactomorphism and order, despite non-unique sphere systems in the underlying topological decomposition?
• RQ2 Does Colin’s connected sum construction for tight contact structures preserve uniqueness of the summands under isotopy and contactomorphism?
• RQ3 To what extent does the uniqueness of the decomposition depend on the tightness of the contact structures and the classification of tight structures on the 3-ball?
• RQ4 How can isotopy invariance of contact structures on cobordisms between spheres be used to eliminate non-minimal sphere systems in the decomposition?
• RQ5 Is the uniqueness of the prime decomposition in the contact category strictly limited to tight contact structures, or does it extend to overtwisted ones?

Key findings

• Every non-trivial tight contact 3-manifold admits a connected sum decomposition into finitely many prime tight contact 3-manifolds.
• The prime tight contact summands in such a decomposition are unique up to order and contactomorphism.
• The connected sum construction for tight contact structures, as defined by Colin and Honda, yields a well-defined contact structure on the sum manifold that is independent of the isotopy class of the gluing spheres.
• The uniqueness relies crucially on Eliashberg’s result that tight contact structures on the 3-ball are uniquely determined by their boundary characteristic foliation.
• A minimal system of 2-spheres in the decomposition can always be found via isotopy, and any non-minimal system can be simplified without changing the contactomorphism type of the summands.
• The proof shows that modifying a sphere system via isotopy to reduce intersection components leads to contactomorphic summands, thereby ensuring uniqueness.