[Paper Review] Generalizations of planar contact manifolds to higher dimensions

A. C.U. Bahar, John B. Etnyre+1 more|arXiv (Cornell University)|Jun 4, 2020

Geometric and Algebraic Topology44 references4 citations

TL;DR

This paper introduces two generalizations of planar contact manifolds to higher dimensions: iterated planar and projective contact manifolds. It establishes that any finitely presented group can be realized as the fundamental group of an iterated planar contact manifold in odd dimensions >3, and proves the existence of symplectic caps for certain projective contact 5-manifolds via symplectic cobordisms and Lefschetz fibrations.

ABSTRACT

Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions. We study some basic topological properties of iterated planar contact manifolds and discuss several examples and constructions showing that many contact manifolds are iterated planar. We also observe that for any odd integer m > 3, any finitely presented group can be realized as the fundamental group of some iterated planar contact manifold of dimension m. Moreover, we introduce another generalization of three dimensional planar contact manifolds that we call projective. Finally, building symplectic cobordisms via open books, we show that some projective contact manifolds admit explicit symplectic caps.

Motivation & Objective

• To extend the theory of planar contact manifolds from 3D to higher odd dimensions using iterated planar and projective open books.
• To investigate the topological and symplectic properties of these generalized contact manifolds.
• To determine whether fundamental group constraints exist for iterated planar contact manifolds.
• To construct explicit symplectic caps for certain projective contact 5-manifolds using symplectic cobordisms and Lefschetz fibrations.

Proposed method

• Define iterated planar contact manifolds via iterated planar Lefschetz fibrations with planar base fibers.
• Use open book decompositions with symplectic pages to support contact structures and analyze their monodromy.
• Construct symplectic cobordisms by attaching handles (X × D²) to the boundary of a symplectic filling, preserving Liouville vector fields.
• Apply rounding of corners to ensure compatibility of the Liouville field with the cobordism structure.
• Utilize symplectic isotopy results (e.g., Li-Wu, Li-Li-Wu) to relate monodromy maps to compositions of Dehn twists on Lagrangian spheres.
• Construct symplectic Lefschetz fibrations over D² with specified monodromy to glue to existing cobordisms and form symplectic caps.

Experimental results

Research questions

• RQ1 Can any finitely presented group be realized as the fundamental group of an iterated planar contact manifold in dimension m > 3?
• RQ2 Do symplectic fillings of iterated planar contact manifolds always have connected boundary?
• RQ3 Which projective contact 5-manifolds admit explicit symplectic caps?
• RQ4 Can the monodromy of a symplectic F-bundle over S¹ (F = CP²#nCP², n ≤ 4) be symplectically isotopic to a composition of Dehn twists?
• RQ5 Under what conditions can a symplectic cobordism be extended to a symplectic cap via Lefschetz fibration attachments?

Key findings

• For any odd dimension m > 3 and any finitely presented group G, there exists an iterated planar contact (2n+1)-manifold with fundamental group G.
• Any symplectic filling of an iterated planar contact manifold must have connected boundary, generalizing a 3D result.
• An iterated planar contact 5-manifold is projective, meaning its Weinstein page embeds as a convex domain in CP²#kCP² for some k.
• If the page of a projective open book embeds in CP²#nCP² with n ≤ 4, then the contact 5-manifold admits a symplectic cap containing CP²#nCP².
• The monodromy of a symplectic F-bundle over S¹ (F = CP²#nCP², n ≤ 4) that acts trivially on homology is symplectically isotopic to the identity.
• By attaching symplectic 3-handles along Lagrangian spheres and using Lefschetz fibrations, a symplectic cobordism can be extended to a symplectic cap for the contact manifold.