[Paper Review] H-principle for 4-dimensional contact foliations

Roger Casals, Álvaro del Pino+1 more|arXiv (Cornell University)|Jun 27, 2014

Geometric and Algebraic Topology14 references3 citations

TL;DR

This paper establishes a parametric existence h-principle for codimension-1 contact foliations on closed, oriented 4-manifolds, proving that any foliated almost contact structure can be homotoped through almost contact structures to a contact foliation. The key technique involves a vanishing family of Lutz twists along transverse intervals in local models, leveraging overtwisted contact structures and foliation-specific geometric flexibility to achieve the result in the 4-dimensional setting.

ABSTRACT

In this article we introduce the topological study of codimension-1 foliations which admit contact or symplectic structures on the leaves. A parametric existence h-principle for foliated contact structures is provided for any cooriented foliation in a closed oriented 4-fold.

Motivation & Objective

• To establish a foundational framework for foliated contact and symplectic topology in codimension-1 foliations.
• To investigate the existence of contact structures on the leaves of a codimension-1 foliation in 4-manifolds.
• To determine whether the existence of a foliated contact structure is governed by topological obstructions rather than geometric ones.
• To prove a parametric h-principle for foliated contact structures in 4-dimensional manifolds.

Proposed method

• The authors define the space of codimension-2 distributions ξ contained in the foliation Tℱ, and introduce the spaces ℂ(𝑉,ℱ) of contact foliations and 𝒜(𝑉,ℱ) of almost contact foliations.
• They use the classification of overtwisted contact structures in 3-manifolds as a key ingredient in the construction.
• A local model is constructed via embeddings κγ: I₂ × D³ → V, where the almost contact structure is made leafwise contact using a contact form dz + r²dθ.
• A vanishing family of Lutz twists is defined along intervals I₂, using a cutoff function χ(t) to interpolate between the original almost contact structure and a twisted one, ensuring the result is foliated contact.
• The construction is extended parametrically over a compact parameter space P by ensuring all embeddings and deformations depend continuously on the parameter.
• The proof applies a relative h-principle in the 4-cell relative to the 3-skeleton, using Theorem 27 to extend the contact structure globally.

Experimental results

Research questions

• RQ1 Can every foliated almost contact structure on a closed, oriented 4-manifold be homotoped to a foliated contact structure?
• RQ2 Is the existence of a foliated contact structure on a codimension-1 foliation strictly a topological question, with no geometric obstructions?
• RQ3 Does the h-principle for contact structures extend to the setting of foliated structures in 4 dimensions?
• RQ4 Can the parametric version of the h-principle be established for foliated contact structures?
• RQ5 What role do overtwisted disks and Lutz twists play in the construction of foliated contact structures?

Key findings

• The paper proves a parametric h-principle for foliated contact structures in 4-dimensional manifolds: any foliated almost contact structure is homotopic through almost contact structures to a contact foliation.
• The space of contact foliations ℂ(𝑉,ℱ) has the same homotopy groups as the space of almost contact foliations 𝒜(𝑉,ℱ), meaning the inclusion ℂ(𝑉,ℱ) → 𝒜(𝑉,ℱ) induces surjections on all homotopy groups.
• The vanishing family of Lutz twists provides a new geometric tool to construct foliated contact structures by deforming almost contact structures in a controlled, compactly supported way along transverse intervals.
• The method relies on a local model where the contact structure is realized as the kernel of dz + r²dθ in I₂ × D³, enabling explicit control over the deformation.
• The result holds for both closed 4-manifolds and open 4-manifolds, and the argument can be adapted to higher codimension foliations with appropriate modifications.
• The map π₀ι: π₀ℂ(𝕊(ℱ), π*ℱ) → π₀𝒜(𝕊(ℱ), π*ℱ) is not injective, showing that foliated contact structures exhibit geometric rigidity in certain cases, contrasting with the flexibility of the h-principle.