[Paper Review] Notes on the isotopy finiteness

Vincent Colin, Emmanuel Giroux+1 more|arXiv (Cornell University)|May 14, 2003

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper establishes that every closed, oriented, atoroidal 3-manifold admits only finitely many isotopy classes of tight contact structures. Using branched surfaces and weight functions to encode twisting numbers along Legendrian fibers, the authors prove finiteness by showing that infinite families of tight contact structures would force the manifold to be toroidal—contradicting the atoroidal assumption.

ABSTRACT

This is the less official, English version of the proof of the fact that every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.

Motivation & Objective

• To prove that closed, oriented, atoroidal 3-manifolds carry only finitely many isotopy classes of tight contact structures.
• To establish a finiteness result in contact topology by analyzing the structure of tight contact structures via branched surfaces.
• To show that the existence of infinitely many tight contact structures would imply the manifold is toroidal, contradicting the atoroidal hypothesis.
• To develop a weight function formalism on branched surfaces to classify tight contact structures up to isotopy.
• To use the Farey tessellation and boundary slope analysis to rule out infinite families of tight contact structures in the atoroidal case.

Proposed method

• Construct a finite collection of branched surface pairs $(\mathcal{B}_i, \zeta_i)$ such that every tight contact structure on $V$ is isotopic to one generated by some pair.
• Assign to each tight contact structure a weight function $w: \mathcal{B} \setminus L \to \mathbb{Z}$, constant on sectors, satisfying branch equations $w(B_1) + w(B_2) = w(B_3)$ at branch loci.
• Use the contractibility of $\mathrm{Diff}^+(I)$ to show that weight functions uniquely determine contact structures up to isotopy relative to $V \setminus N(\mathcal{B})$.
• Analyze boundary slopes of Seifert fibered pieces in a JSJ decomposition, particularly in the case of three singular fibers.
• Apply Farey tessellation and rational slope analysis to show that infinite families of contact structures would require $\sum_{i=1}^3 \frac{1}{\alpha_i} = 1$, implying the manifold is a torus bundle.
• Derive a contradiction by showing that such a condition forces the manifold to be toroidal, violating the atoroidal assumption.

Experimental results

Research questions

• RQ1 Can the number of isotopy classes of tight contact structures on a closed, atoroidal 3-manifold be finite?
• RQ2 What topological constraints arise if infinitely many tight contact structures exist on such a manifold?
• RQ3 How do weight functions on branched surfaces classify tight contact structures up to isotopy?
• RQ4 Under what conditions does a Seifert fibered space with three singular fibers admit infinitely many tight contact structures?
• RQ5 What role does the Farey tessellation play in detecting maximal twisting and ruling out infinite families of contact structures?

Key findings

• Every closed, oriented, atoroidal 3-manifold admits only finitely many isotopy classes of tight contact structures.
• The existence of infinitely many tight contact structures would force the manifold to be a torus bundle over the circle, contradicting the atoroidal hypothesis.
• Weight functions on branched surfaces, satisfying the branch equations $w(B_1) + w(B_2) = w(B_3)$, uniquely determine tight contact structures up to isotopy.
• The condition $\sum_{i=1}^3 \frac{1}{\alpha_i} = 1$ characterizes Seifert fibered spaces that could potentially admit infinitely many tight contact structures.
• For large $k$, the boundary slope $s_k$ of the third piece lies strictly above $\frac{\beta_3}{\alpha_3}$ and is connected to it by an edge in the Farey tessellation, ensuring maximality of twisting.
• The determinant condition $\det(\alpha_3, \beta_3; \text{numerator}, \text{denominator}) = 1$ implies $\sum_{i=1}^3 \frac{1}{\alpha_i} = 1$, which characterizes torus bundles over $S^1$.