[Paper Review] Algebraic Torsion in Higher-Dimensional Contact Manifolds

Agustín Moreno|arXiv (Cornell University)|Apr 4, 2019

Geometric and Algebraic Topology39 references10 citations

TL;DR

This paper constructs the first known examples of higher-dimensional contact manifolds (in all odd dimensions ≥5) with finite, non-zero algebraic torsion, proving they are tight and admit no strong symplectic fillings. It establishes that Giroux torsion implies algebraic 1-torsion in any odd dimension, resolves a conjecture from [MNW13], and constructs infinitely many non-diffeomorphic 5-manifolds with no Giroux torsion yet still non-fillable. A novel intersection theory for punctured holomorphic curves generalizes 3D results to higher dimensions, enabling control over curve behavior via codimension-2 holomorphic foliations.

ABSTRACT

Wir konstruieren Beispiele von Kontaktmannigfaltigkeiten in jeder ungeraden Dimension, welche endliche nicht-triviale algebraische Torsion (im Sinne von Latschev-Wendl) aufweisen, somit straff sind und keine starke symplektische Füllung haben. Wir beweisen, dass Giroux Torsion algebraische 1-Torsion in jeder ungeraden Dimension impliziert, womit eine Vermutung von Massot-Niederkrüger-Wendl bewiesen wird. Wir konstruieren unendlich viele nicht diffeomorphe Beispiele von 5-dimensionalen Kontaktmannigfaltigkeiten, welche straff sind, keine starke symplektische Füllung zulassen und keine Giroux Torsion haben. Wir erhalten Obstruktionen für symplektische Kobordismen, ohne für deren Beweis die SFT Maschinerie zu verwenden. Wir geben eine provisorische Definition eines spinalen offenen Buchs in höherer Dimension an, basierend auf der vom 3-dimensionalen Fall aus Lisi-van Horn Morris-Wendl. In einem Anhang geben wir in gemeinsamer Autorenschaft mit Richard Siefring eine wesentliche Zusammenfassung der Schnitttheorie für punktierte holomorphe Kurven und Hyperflächen an, welche die 3-dimensionalen Resultate von Siefring auf höhere Dimensionen verallgemeinert. Mittels der Schnitttheorie erhalten wir eine Anwendung für holomorphe Blätterungen von Kodimension zwei, die wir benutzen um das Verhalten von holomorphem Kurven in unseren Beispielen einzuschränken.

Motivation & Objective

• To construct contact manifolds in all odd dimensions with finite, non-zero algebraic torsion, proving they are tight and non-fillable.
• To prove that Giroux torsion implies algebraic 1-torsion in any odd dimension, resolving a conjecture from [MNW13].
• To construct infinitely many non-diffeomorphic 5-dimensional contact manifolds that are tight, non-fillable, and lack Giroux torsion.
• To develop a higher-dimensional generalization of the intersection theory for punctured holomorphic curves and hypersurfaces, extending results from [Sie11].
• To provide a non-SFT proof of obstruction results for symplectic cobordisms and to define a tentative higher-dimensional spinal open book decomposition.

Proposed method

• Constructing model A via a cylindrical Liouville semi-filling and deforming to a contact structure with finite algebraic torsion.
• Using a finite energy foliation and index computations to control holomorphic curve behavior in the symplectization.
• Applying a perturbation scheme to transition from Morse–Bott to Morse data, ensuring Fredholm regularity of holomorphic curves.
• Introducing a higher-dimensional intersection theory for punctured holomorphic curves and hypersurfaces, based on asymptotic normal behavior and Conley–Zehnder indices.
• Using the intersection theory to restrict holomorphic curves to leaves of codimension-2 holomorphic foliations, particularly in model B and 5D examples.
• Defining a higher-dimensional spinal open book decomposition based on the 3D model from [L-VHM-W], and proving obstruction results via perturbation and regularity techniques.

Experimental results

Research questions

• RQ1 Do there exist higher-dimensional contact manifolds with finite, non-zero algebraic torsion that are tight and non-fillable?
• RQ2 Does Giroux torsion imply algebraic 1-torsion in any odd dimension, as conjectured in [MNW13]?
• RQ3 Can one construct infinitely many non-diffeomorphic 5-dimensional contact manifolds that are tight, non-fillable, and lack Giroux torsion?
• RQ4 What is the behavior of holomorphic curves in higher-dimensional contact manifolds with algebraic torsion, and can it be controlled via foliation theory?
• RQ5 Can obstruction results for symplectic cobordisms be proven without relying on SFT machinery?

Key findings

• The paper constructs, for every odd dimension ≥5, contact manifolds with finite, non-zero algebraic torsion, which are therefore tight and do not admit strong symplectic fillings.
• It proves that Giroux torsion implies algebraic 1-torsion in any odd dimension, confirming a conjecture from [MNW13].
• It constructs infinitely many non-diffeomorphic 5-dimensional contact manifolds that are tight, admit no strong symplectic fillings, and have no Giroux torsion.
• A new intersection theory for punctured holomorphic curves and hypersurfaces is developed, generalizing [Sie11] to higher dimensions and allowing control over curve behavior via codimension-2 holomorphic foliations.
• The theory shows that under certain normal Conley–Zehnder index bounds, holomorphic curves must lie entirely within a leaf of the foliation if the degree of the projection map is zero.
• The paper provides a non-SFT proof of obstruction results for symplectic cobordisms, relying on perturbation, regularity, and the new intersection theory.