Foliations and the topology of 3-manifolds. (English) Zbl 0533.57013

Let M be a compact, connected, oriented 3-manifold with boundary. It is known that, if M admits a transversely oriented foliation by surfaces, transverse to \(\partial M\) and having no Reeb components, then \(\partial M\) is a union (possibly empty) of tori and either M is irreducible or \(M=S^ 1\times S^ 2\) (with the product foliation). Assume that \(H_ 2(M,\partial M)\neq 0\). By work of Thurston, any compact leaf L of such a foliation minimizes the Thurston norm of \([L]\in H_ 2(M,\partial M)\). The author proves a theorem (Theorem 5.5) which, in particular, says that when M satisfies the above necessary conditions, any norm minimizing surface can be realized as a compact leaf of a foliation, transverse to \(\partial M\) and without Reeb components. There are many interesting corollaries, including an answer to the Reeb placement problem of Laudenbach and Roussarie, proofs of two conjectures of Thurston, and a new proof of Dehn’s lemma for higher genus surfaces.
Reviewer: L.Conlon


57R30 Foliations in differential topology; geometric theory
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M35 Dehn’s lemma, sphere theorem, loop theorem, asphericity (MSC2010)
57R19 Algebraic topology on manifolds and differential topology
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