Foliations and the topology of 3-manifolds. II. (English) Zbl 0627.57012

Gabai, David

J. Differ. Geom. 26, 461-478 (1987).

In this paper the author gives some new applications of the technique introduced in the first part of this paper [ibid. 18, 445-503 (1983; Zbl 0533.57013)]. So the basic result is again a theorem about the existence of nice foliations on a 3-manifold having a given Thurston norm minimizing surface as a leaf. This theorem is applied to a question on knots in 3-manifolds.
In particular, three old problems of J. Martin [R. Kirby, Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math. 32, Pt. 2, 273-312 (1978; Zbl 0394.57002), Problems 1.18 A,B,C] are solved. The type of application is properly illustrated by the following result (the solution of one of Martin’s problems; Corollary 2.5): Let k be a knot in \(D^ 2\times S^ 1\) of winding number 0 such that k does not lie in a 3-cell in \(D^ 2\times S^ 1\). If M is obtained by nontrivial surgery on k, then \(M\neq D^ 2\times S^ 1\).

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)

Keywords:

foliations on a 3-manifold; Thurston norm minimizing surface; knots in 3- manifolds; nontrivial surgery

Citations:

Zbl 0533.57013; Zbl 0394.57002