Foliations and the topology of 3-manifolds

Corollary 6. 5. A nontήvial link Lin S 3 is nonsplit if and only ifL is the set of cores of Reeb components of some foliation ΦofS 3. The => direction follows from Theorem 5. 5. Novikov 21 proved the converse in 1965. We therefore answer the so-called "Reeb placement problem" of Laudenbach and Roussarie 16 who asked which links could be realized as cores of Reeb components of foliations of S 3. The holonomy of our foliations along the toral leaves is in general C°. The C 00 problem is open although it can be solved for the alternating knots, fibred knots, many other knots, and certain "sums" of such knots using the constructions 6-8 Corollary 6. 7. Let R t be a Seifert surface for the oriented link L, C S 3 for i -1, 2, and R be any Murasugi sum (or generalized plumbing) of R { and R 2 with L -dR. Then R is a minimal genus surface for the oriented link L if and only if each /^ is a minimal genus surface for the oriented link L z. This generalizes the classical result due to Seifert in the 1930's that the connected sum of minimal genus surfaces is a surface of minimal genus. Corollary 6. 9. Let M be a compact connected irreducible oriented 3-manifold whose boundary ΘM is a (possibly empty) union of incompressible tori, and H 2 (M, dM) is not generated by tori and annuli. Then there exists a C 00 transversely oriented foliation ^ on M such that ^ ίίl ΘM, < \ ΘM has no Reeb components, and no leaf of ^ is compact. In particular we have Corollary 6. 11. Let M be either a compact connected oriented 3-manifold whose interior has a complete hyperbolic metric and H 2 (M, ΘM) Φ 0, or M -S 3 -N (L) where L is a nonsplit nontrivial link in S 3. Then there exists a C°°t ransversely oriented foliation ¥ of M such that 3F has no compact leaves, ®ί ffl ΘM, and < \ ΘM has no Reeb components. The conditions that ΘM be a union of incompressible tori and M be irreducible are necessary by Novikov's work. The question of whether a manifold possesses a C°° codimension-1 foliation without compact leaves has been precisely answered by the work of Thurston 31, Levitt 18, Wood 34, and Milnor 19 for circle bundles over surfaces and for most Seifert fibred spaces by 4; see also 5. The 2-dimensional homology of these spaces (except for trivial cases) is generated by tori and annuli. It would be interesting to