Foliations and the topology of 3-manifolds. III

DAVID GABAI Corollary 8.6 (Poenaru 1974).// V is a 4-manifold obtained by attaching a 2-handle and a 3-handle to B 4 such that H 2 (V) = 0, then V = B 4 .Remark (September 1986).M. Scharlemann has found a simplification of our proof of Corollary 8.3 which avoids foliations (see Remark 8.3^).For ten ways to compute the genus of a knot in S 3 consult Theorem 8.8.The smoothing procedure of [3, §5] allows one to modify the construction of the foliation of Theorem 3.1 to obtain a smooth one if genus k Φ 1. Applying a further modification we can eliminate the compact leaves.In the case that genus k = 1 these modifications may throw holonomy onto the boundary.Therefore we obtain the following result.Corollary 8.10.// k is a knot in S 3 such that genus A: > 1, and S is a minimal genus Seifert surface for k, then there exists C 00 , taut foliations J^ , / = 1, 2 of S 3 -N(k) such that ^ (k) is a foliation by circles, S is a leaf of J^, and no leaf of J^2 is compact.Tubularizing these foliations near dN(k) and attaching a Reeb component yields Corollary 8.11.A C° version of this result (for k nontrivial) was obtained in [3].Corollary 8.11.Ifk is a knot in S 3 such that genus A: > 1, then there exists a C 00 foliation IF of S 3 with a single Reeb component whose core is isotopic to k.The most striking observation in the proof of Theorem 3.1 is that any nice finite depth taut partial foliation constructed on S 3 -N(k) extends to a foliation satisfying the conclusions of that result.This is the key ingredient in proving Corollary 8.19.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 ι .Since the trefoil and the figure 8 knots are the only genus one fibered knots in S 3 [8] we obtain Corollary 8.23.Surgery on a knot in S 3 yields a torus bundle over S ι 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.We assume that the reader is familiar with the results and terminology of [3] and of §0 of [6].§1 and §2 of [6] are independent of this paper.Because we view this paper as a continuation of Foliations and the topology of 3-manifolds II, we begin with §3.The proof of Theorem 3.1 involves four steps which are respectively carried out in § §3-6.These four steps are precisely stated and put together in §7.The reader is advised to consult §7 for an overview of the proof.Consequences of Theorem 3.1 are given in §8.The author gratefully thanks M. Scharlemann and T. Kobayashi for their large number of constructive criticisms of the text.