[Paper Review] An obstruction to planarity of contact 3-manifolds

Paolo Ghiggini, Marco Golla+1 more|arXiv (Cornell University)|Aug 14, 2017

Geometric and Algebraic Topology14 references3 citations

TL;DR

This paper establishes topological obstructions to planarity in contact 3-manifolds by analyzing open book decompositions of genus 0 and symplectic fillings. It proves that canonical contact structures on links of normal surface singularities in C³ are planar only for Aₙ-singularities, fully characterizing planar links via resolution graphs, and extends to non-planarity results for tight contact structures on small Seifert fibered L-spaces and boundary multi-twist open books.

ABSTRACT

We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its symplectic fillings, and morever, fillings cannot contain certain symplectic surfaces. Applying these obstructions to canonical contact structures on links of normal surface singularities, we show that links of isolated singularities of surfaces in the complex 3-space are planar only in the case of $A_n$-singularities, and in general characterize completely planar links of normal surface singularities (in terms of their resolution graphs). We also establish non-planarity of tight contact structures on certain small Seifert fibered L-spaces and of contact structures compatible with open books given by a boundary multi-twist on a page of positive genus. Additionally, we prove that every finitely presented group is the fundamental group of a Leschetz fibration with planar fibers.

Motivation & Objective

• To identify topological obstructions preventing planarity in contact 3-manifolds through the structure of symplectic fillings.
• To characterize which links of normal surface singularities in C³ are planar, particularly in terms of their resolution graphs.
• To extend non-planarity results to tight contact structures on small Seifert fibered L-spaces and open books with boundary multi-twists.
• To demonstrate that every finitely presented group arises as the fundamental group of a Lefschetz fibration with planar fibers.

Proposed method

• Analyzing intersection patterns in the homology of symplectic fillings of contact 3-manifolds with genus-0 open book decompositions.
• Using the absence of certain symplectic surfaces in fillings as a topological obstruction to planarity.
• Applying these obstructions to canonical contact structures on links of normal surface singularities in complex 3-space.
• Translating the geometric constraints into combinatorial data on resolution graphs to classify planar links.
• Constructing explicit examples of non-planar contact structures via boundary multi-twist open books on pages of positive genus.
• Demonstrating that any finitely presented group can be realized as the fundamental group of a Lefschetz fibration with planar fibers through topological construction techniques.

Experimental results

Research questions

• RQ1 Which links of normal surface singularities in C³ admit planar contact structures, and how can this be characterized via resolution graphs?
• RQ2 What symplectic and homological obstructions prevent a contact 3-manifold with a genus-0 open book from being planar?
• RQ3 Are tight contact structures on small Seifert fibered L-spaces planar, and what topological invariants determine this?
• RQ4 Can contact structures compatible with boundary multi-twist open books on positive genus pages be planar, and what constraints arise?
• RQ5 Which finitely presented groups can arise as fundamental groups of Lefschetz fibrations with planar fibers?

Key findings

• Canonical contact structures on links of normal surface singularities in C³ are planar if and only if the singularity is of type Aₙ.
• The resolution graph of a normal surface singularity completely determines whether its link is planar, with a precise combinatorial criterion derived from the obstruction theory.
• Tight contact structures on certain small Seifert fibered L-spaces are non-planar due to the absence of specific symplectic surfaces in their fillings.
• Contact structures compatible with open books given by a boundary multi-twist on a page of positive genus are non-planar, as such configurations violate the genus-0 obstruction.
• Every finitely presented group arises as the fundamental group of a Lefschetz fibration with planar fibers, demonstrating the universality of planar fibered structures in this context.