Low-Dimensional Topology

Manifolds are central objects of study in both geometry and topology. Our group focuses primarily on manifolds that are three-dimensional. These are spaces that locally resemble the standard three-dimensional Euclidean space, but their global structure can be much more complicated.

A simple way to construct a nontrivial three-manifold is to take the Cartesian product of a closed interval with a surface (two-dimensional manifold), such as the two-sphere (the surface of the Earth), the two-torus (the surface of a donut), or a surface of higher genus. This is a three-manifold with two boundary components, each homeomorphic to the chosen surface. Identifying the points in one component with the points in the other component via some homeomorphism yields a closed three-manifold. Manifolds obtained in this way are called mapping tori of surface homeomorphisms, or three-manifolds fibered over the circle. Not all closed three-manifolds fiber over the circle. However, if a manifold is hyperbolic then it has a finite cover that does.

Every three-manifold that is fibered over the circle admits the suspension flow, the unit speed flow that moves points transversely to the fibers. Under certain assumptions on the topology of the fiber and the gluing homeomorphism (monodromy), one can ensure that this flow is pseudo-Anosov. This means that the manifold admits a pair of transverse singular foliations, each foliated by flow lines of the flow, and such that all flow lines within a single leaf are forward asymptotic in one foliation and backward asymptotic in the other.

Pseudo-Anosov flows have very interesting dynamical properties, deep connections with the geometry and topology of the underlying manifold, its foliations and laminations, contact structures, and properties of its fundamental group. They also exist on non-fibered three-manifolds. Recently, it was shown that pseudo-Anosov flows can be encoded by finite combinatorial structures called veering triangulations. Since then, several new results on pseudo-Anosov flows have been obtained through the use of veering triangulations.

Much of our group’s work revolves around pseudo-Anosov flows and veering triangulations. Among the directions we are investigating are:

• How many distinct veering triangulations/pseudo-Anosov flows does a fixed three-manifold admit?
• How do algebraic invariants of the manifold inform about pseudo-Anosov flows/veering triangulations of that manifold?
• Relationships between distinct pseudo-Anosov flows/veering triangulations on the same manifold.
• Recognizing different veering triangulations encoding the same flow.
• Combinatorial operations on veering triangulations and underlying operations on pseudo-Anosov flows.
• Veering triangulations and hyperbolic geometry.

Other topics in three-manifold topology, as well as generalizations to higher dimensions, are also of interest.