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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

##### MSC:
 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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