Introduction to taut foliations on three-manifolds

Hiraku Nozawa (Shiga)

Resumo:

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.

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© Hiraku Nozawa.