Foliations and the topology of 3-manifolds. III.

*(English)*Zbl 0639.57008[For parts I, II see ibid. 18, 445-503 (1983; Zbl 0533.57013), 26, 461- 478 (1987; Zbl 0627.57012).]

As in parts I, II, the author first proves a theorem about the existence of some nice foliations on 3-manifolds and then gives several remarkable applications to the classical 3-manifold topology. The main theorem is the following: If S is a minimal genus Seifert surface for a knot k in S 3, then there exists a taut finite depth foliation F of S 3-\(\overset\circ N(k)\), where N(k) is a tubular neighborhood of k, such that S is a leaf of F an \(F| \partial N(k)\) is a foliation by circles. One of the applications is the following result: If M is obtained by performing zero frame surgery on a knot k in S 3, then M is prime and genus k\(=\min \{genus S|\) S is a nonseparating oriented embedded surface in \(M\}\). This gives, in particular, a proof of two important conjectures: the property R conjecture and the Poenaru conjecture. The author notes that one can eliminate the foliations from the proof of this result. This remark is due to M. Scharlemann. But the technique of satured manifolds developed in this series of papers to construct foliations remains as the main ingredient of the proof. Another application gives ten ways to compute the genus of a knot in S 3. Note also the following nice corollary: k is a fibered knot in S 3 if and only if the manifold M obtained by performing zero frame surgery to k fibers over S 1. The last corollary of this remarkable paper is the following: Surgery on a knot in S 3 yields a torus bundle over S 1 if and only if the surgery is the zero frame one and either k is the trefoil knot or k is the figure 8 knot. The reader should not miss other papers by the author [see, e.g., Topology 23, 381-394 (1984; Zbl 0567.57021); Mem. Am. Math. Soc. 339, 1-98 (1986; Zbl 0585.57003)].

As in parts I, II, the author first proves a theorem about the existence of some nice foliations on 3-manifolds and then gives several remarkable applications to the classical 3-manifold topology. The main theorem is the following: If S is a minimal genus Seifert surface for a knot k in S 3, then there exists a taut finite depth foliation F of S 3-\(\overset\circ N(k)\), where N(k) is a tubular neighborhood of k, such that S is a leaf of F an \(F| \partial N(k)\) is a foliation by circles. One of the applications is the following result: If M is obtained by performing zero frame surgery on a knot k in S 3, then M is prime and genus k\(=\min \{genus S|\) S is a nonseparating oriented embedded surface in \(M\}\). This gives, in particular, a proof of two important conjectures: the property R conjecture and the Poenaru conjecture. The author notes that one can eliminate the foliations from the proof of this result. This remark is due to M. Scharlemann. But the technique of satured manifolds developed in this series of papers to construct foliations remains as the main ingredient of the proof. Another application gives ten ways to compute the genus of a knot in S 3. Note also the following nice corollary: k is a fibered knot in S 3 if and only if the manifold M obtained by performing zero frame surgery to k fibers over S 1. The last corollary of this remarkable paper is the following: Surgery on a knot in S 3 yields a torus bundle over S 1 if and only if the surgery is the zero frame one and either k is the trefoil knot or k is the figure 8 knot. The reader should not miss other papers by the author [see, e.g., Topology 23, 381-394 (1984; Zbl 0567.57021); Mem. Am. Math. Soc. 339, 1-98 (1986; Zbl 0585.57003)].

Reviewer: N.Ivanov

##### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57R30 | Foliations in differential topology; geometric theory |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |