Introduction to taut foliations on three-manifolds

Hiraku Nozawa (Shiga)

Venres 15 e sábado 16 de xuño de 2018 ás 12 horas na aula 7, como parte do "Advanced Course on Geometry, Topology and Global Analysis of Foliated Spaces" organizado por Jesús A. Álvarez López, Enrique Macías Virgós e Antonio Gómez Tato.

A codimension q foliation on a n-manifold M is a decomposition of M into (n-q)-submanifolds which is trivial on an neighborhood of each point of M. The geometry of codimension one foliations on 3-manifolds has been extensively studied with its relation to topology and dynamics in dimension 3. The tautness is crucial in this theme: A codimension one foliation F on a closed manifold M is taut if for every leaf of F, there exists a transversal loop of F which intersect L. By Sullivan's theorem, tautness of F is equivalent to the existence of a Riemannian metric g on M such that every leaf of F is a minimal submanifold of (M, g).

Taut foliations on 3-manifolds have a mysterious finite aspect. Here is a short list of known results:

- Lickorish proved that any closed 3-manifold admits a codimension one foliation

- On the other hand, Novikov and Rosenberg showed several necessary topological conditions for 3-manifols M to admit a taut foliation, e.g., the fundamental group should be infinite.

- Kronheimer-Mrovka proved that the homotopy classes of tangent plane fields of taut foliations are finite.

- Ghys-Sergiescu proved that torus bundles with hyperbolic monodromy have two foliations without compact leaves up to isotopy.

- Gabai, Cantwell-Conlon, Roberts, Li, Tejas, Brittenham, Jun, Nakae constructed taut foliations on many 3-manifolds with boundary like as knot complements.

After reviewing these classical results, we will propose several problems for the classification of taut foliations on hyperbolic 3-manifolds by describing various examples of taut foliations on surface bundles over circle due to Cooper-Long-Reid, Matsumoto, Meigniez,Alcalde Cuesta-Dal'bo-Martínez-Verjovsky, and the author and hector.

© Hiraku Nozawa.