Essential laminations in surgered $3$-manifolds

Ying Qing Wu
1992 Proceedings of the American Mathematical Society  
In generic cases, an essential lamination in the interior of a 3manifold will remain essential after most of the Dehn fillings along a torus boundary component. Suppose AZ is a 3-manifold with torus F asa boundary component, and let P be an incompressible surface on <9AZ disjoint from T. It was proved in [9] that in most cases, P remains incompressible in most of the Dehn filled manifolds M(y). (See Proposition 2 below.) The present note is to solve a problem posed informally by Peter Shalen,
more » ... ich asks whether a similar result holds for essential laminations in 3-manifolds. The answer is positive: Essential laminations disjoint from T will usually remain essential after Dehn fillings. Note that the essentiality of a lamination concerns not only compressibility but also reducibility and existence of end compressing disks in the resulting manifolds. Throughout this paper, all 3-manifolds are assumed orientable. Let AZ be a 3-manifold and T a torus component of the boundary of AZ. For a slope y on T, let M(y) be the manifold obtained by attaching a solid torus V to T such that y corresponds to the meridian slope of V. If yx, y2 are two slopes on T, denote their geometric intersection number by A = A(j>i, y2). We refer the readers to [4] for the definition of essential laminations and related notions like branched surfaces, horizontal boundary, vertical boundary, monogons, etc. Now suppose M is compact, and let X be an essential lamination in AZ. We assume that X is disjoint from ÖAZ. A set A is called a quasi-annulus if A is the image of a continuous map /: Sx x I -► AZ, such that f\s¡ X[o, i) is an embedding, f(Sx x 0) = A n T, and f(Sx x 1) = A n X. A is incompressible if f(Sx x [0, 1)) is incompressible. Theorem 1. Suppose M contains no incompressible quasi-annulus from T to X. If X is not essential in M(yx) and M(y2), then A(yx, y2) < 1. Thus X is essential in M(y) for all but at most three slopes y. Remark. Generally, the quasi-annulus in the theorem cannot be replaced by an embedded annulus. For example, let A be a Klein bottle, and let Mx be a twisted I-bundle over X. Let AZ2 be a (p, q) cable space, q > 1. (See e.g.,
doi:10.1090/s0002-9939-1992-1104405-0 fatcat:pqhortmtmjgqzbukvzu3klqkpa