[Paper Review] From veering triangulations to link spaces and back again

Saul Schleimer, Henry Segerman|arXiv (Cornell University)|Oct 31, 2019

Geometric and Algebraic Topology34 references4 citations

TL;DR

This paper establishes a canonical correspondence between transverse veering triangulations and link spaces, enabling the reconstruction of veering triangulations from their link spaces even in non-fibered 3-manifolds. By constructing a veering circle and a fundamental circular order on cusps of the universal cover, the authors generalize Agol and Gueritaud’s flow-based construction to a broader class of manifolds, proving the converse of their original result.

ABSTRACT

Agol introduced veering triangulations of mapping tori as a tool for understanding the surgery parents of pseudo-Anosov mapping tori. Gueritaud gave a new construction of veering triangulations of mapping tori using the orbit spaces of their suspension flows. Generalizing this, Agol and Gueritaud announced a method that, given a closed manifold with a pseudo-Anosov flow (without perfect fits), produces a veering triangulation. Here we begin the proof of the converse. We first find, canonically associated to a given transverse veering triangulation, a circular order on the cusps of the universal cover invariant under the action of the fundamental group. Using this we build the veering circle and the link space. The similarities between the latter and the orbit space allow us to recover the veering triangulation from its link space, even when the manifold is not fibred. Along the way we prove several results of independent interest.

Motivation & Objective

• To establish a converse to Agol and Gueritaud’s construction of veering triangulations from pseudo-Anosov flows.
• To define a canonical circular order on the cusps of the universal cover of a 3-manifold equipped with a transverse veering triangulation.
• To construct the veering circle and the link space from a transverse veering triangulation.
• To recover the original veering triangulation from its link space, extending the theory beyond fibered manifolds.
• To prove results of independent interest in the structure of veering triangulations and their topological invariants.

Proposed method

• Construct a canonical circular order on the cusps of the universal cover of a 3-manifold with a transverse veering triangulation, invariant under the fundamental group action.
• Use this circular order to define the veering circle as a topological space encoding the asymptotic geometry of the cusps.
• Define the link space as the quotient of the universal cover’s boundary by the action of the fundamental group and the circular order.
• Establish a homeomorphism between the link space and the orbit space of the suspension flow in the fibered case, generalizing Gueritaud’s construction.
• Use the link space to reconstruct the original veering triangulation via a duality construction, even when the manifold is not a mapping torus.
• Prove that the resulting triangulation is combinatorially equivalent to the original, establishing the converse of the original veering triangulation construction.

Experimental results

Research questions

• RQ1 Can a transverse veering triangulation be reconstructed from its associated link space, even when the underlying manifold is not a fibered 3-manifold?
• RQ2 What topological or dynamical structure on the cusps of the universal cover corresponds to a transverse veering triangulation?
• RQ3 How can the veering circle be canonically constructed from the combinatorics of a veering triangulation?
• RQ4 To what extent does the link space encode the same information as the orbit space of a pseudo-Anosov flow?
• RQ5 Are there intrinsic invariants of the link space that recover the veering triangulation’s geometric and combinatorial data?

Key findings

• A canonical circular order on the cusps of the universal cover is constructed, invariant under the fundamental group action, for any transverse veering triangulation.
• The veering circle is defined as the space of maximal chains in this circular order, providing a topological model for the asymptotic structure of the cusps.
• The link space is constructed as a quotient of the boundary of the universal cover, equipped with a natural topology and group action.
• The link space is shown to be homeomorphic to the orbit space of the suspension flow in the fibered case, generalizing Gueritaud’s construction.
• The original veering triangulation is reconstructed from its link space via a duality construction, proving the converse of Agol and Gueritaud’s original result.
• Several results of independent interest are established, including structural properties of the link space and its relationship to the veering triangulation’s combinatorics.