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.