Constructing Essential Laminations in Some 3-Manifolds

Citation

Naimi, Ramin (1992) Constructing Essential Laminations in Some 3-Manifolds. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/n8jw-6c75. https://resolver.caltech.edu/CaltechTHESIS:08242011-132722715

Abstract

In trying to understand 3-manifolds (with the hope of eventually classifying them as with 2-manifolds), one approach that has turned out to be fruitful is to study objects of codimension one in them, more specifically, incompressible surfaces, taut foliations (or foliations without Reeb components), and essential laminations (loosely speaking, a lamination in a manifold is a foliation of a closed subset of that manifold). For a 3-manifold M containing an incompressible surface (with some extra hypotheses), Waldhausen proved great theorems such as: M has infinite fundamental group, the universal cover of M is R^3, homotopic homeomorphisms of M are isotopic, and π_1 (M) determines M up to homeomorphism. Similar theorems for manifolds containing taut foliations were proven by Novikov, Haefiiger, Rosenberg, and others. The essential lamination was developed comparatively recently (late 1980's) as a generalization of the incompressible surface and the taut foliation, which themselves qualify as essential laminations. In fact they are just extreme cases of essential laminations: at one end we have surfaces, which are properly embedded, and at the other end we have foliations, which fill up the manifold (empty complement). A typical lamination is in general somewhere in between; it is nowhere dense as in the case of surfaces, but has non-compact leaves as in foliations. Analogues of some of the theorems of Waldhausen have been proven ([GO]) for closed manifolds admitting essential laminations: they are irreducible, have infinite fundamental group, and are covered by R^3. Other questions such as whether homotopic homeomorphisms are isotopic are being worked on. An advantage of the essential lamination over its ancestors, the incompressible surface and the taut foliation, is that it is much more common and easier to find. It is conjectured (or hoped) that "most" closed 3-manifolds admit essential laminations (and the results of this thesis are in support of this conjecture). So believing that the essential lamination is indeed a useful tool in the study of 3-manifolds, and hoping that it is common in them, an important question that arises is: Which 3-manifolds admit essential laminations? In this thesis we answer this question for those manifolds obtained by surgery on 2-bridge knots in the 3-sphere: Theorem 1: Surgery on a 2-bridge torus knot T_(2,q), with coefficient ∈ (-∞, q - 2) yields a manifold which admits essential laminations. Theorem 2: Nontrivial surgery on a non-torus 2-bridge knot yields a manifold which admits essential laminations. This gives as a corollary, for example, that property P is true for 2-bridge knots. The main method used for 2-bridge knots was to start with a given branched surface or lamination on a given knot and go through some Kirby Calculus to see what this lamination looks like on a different knot for which we cannot get all the desired laminations, and then to try to generalize this "newly found" construction. This is quite general and could potentially be as fruitful for all knots and links.

Thesis Files