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Foliations and the topology of 3-manifolds. (English) Zbl 0539.57013

If one tries to foliate a compact 3-manifold by surfaces then, in general, this cannot be done without using Reeb components. Therefore, the question is under what conditions a compact (orientable) 3-manifold M admits a codimension-1 foliation without Reeb components. If \({\mathcal F}\) is such a foliation which, moreover, is transversely orientable and transverse to the boundary \(\partial M\) then it is known [cf. H. Rosenberg, Topology 7, 131-138 (1968; Zbl 0157.305); S. P. Novikov, Tr. Mosk. Mat. Obshch. 14, 248-278 (1965; Zbl 0247.57006)] that either \(M=S^ 1\times S^ 2\) or M is irreducible. Furthermore, if L is a compact leaf of \({\mathcal F}\) then \([L]\in H_ 2(M,\partial M,{\mathbb{R}})\) minimizes the Thurston norm. The main result of this work now states that the above necessary conditions on M for the existence of \({\mathcal F}\) are also sufficient when \(H_ 2(M,\partial M)\neq 0\). More precisely, if S is any norm minimizing surface in M representing a non-trivial element of \(H_ 2(M,\partial M)\) then S can be realized as a leaf of a foliation \({\mathcal F}\) as above. There are several interesting corollaries of this result concerning link complements in \(S^ 3\) as well as foliations without any compact leaf. Also a generalization of a theorem of Roussarie and Thurston concerning the set of tangencies of an immersed surface with respect to (M,\({\mathcal F})\) is announced. Detailed proofs of all these results can be found in the author’s article in J. Differ. Geom. 18, 445- 503 (1983; Zbl 0533.57013).
Reviewer: U.Hirsch

MSC:

57R30 Foliations in differential topology; geometric theory
57N10 Topology of general \(3\)-manifolds (MSC2010)
55N10 Singular homology and cohomology theory
57R95 Realizing cycles by submanifolds
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References:

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