Essential laminations in 3-manifolds.

*(English)*Zbl 0685.57007Haken 3-manifolds are those which contain a two-sided 2-manifold whose fundamental group is a subgroup of the fundamental group of the 3- manifold (i.e. the homomorphism of fundamental groups induced by inclusion is injective). The successful study of this class of 3- manifolds was initiated in the 60’s by Waldhausen, and through work of many people their structure is well-understood. One of Waldhausen’s early results was the fact that the universal cover of a Haken 3-manifold must be homeomorphic to a 3-cell from which some of the boundary has been removed. In particular, if the Haken 3-manifold is closed, then the universal cover is 3-dimensional Euclidean space, rather than one of the many other simply-connected open 3-manifolds which have a more complicated end structure.

Non-Haken 3-manifolds (other than those whose universal cover is the 3- sphere), are still rather poorly understood. By the mid 70’s, there were only a few examples known of non-Haken 3-manifolds, and it was felt that they were probably quite rare. This feeling turned out to be totally incorrect, because Thurston showed how to construct huge quantities of non-Haken 3-manifolds. Thurston’s method shows that “most” ways of attaching solid tori to the torus boundaries of hyperbolic knot and link complements result in non-Haken 3-manifolds.

Currently there is much effort to try to understand non-Haken 3-manifolds by using laminations to play the role that 2-dimensional submanifolds do in the Haken theory. A lamination is a foliation of a closed subset of a manifold, whose leaves have codimension 1 in the manifold. The systematic use of laminations in low-dimensional topology was initiated by Thurston, who introduced the device of “train tracks”, or branched 1-dimensional submanifolds of 2-manifolds. By assigning numbers, called weights, to the tracks in a train track, one can construct the laminations “carried” by the train track. In this way, a train track parametrized a portion of the space of (measured) laminations in the 2-manifold. Laminations in 3- manifolds are studied using the 2-dimensional analogue of train tracks: branched 2-manifolds. Through work of Floyd, Gabai, Hatcher, Morgan, Oertel, Shalen, and others, considerable technical machinery has been developed for working with laminations by using branched 2-manifolds.

In this paper, the authors give two equivalent definitions of a kind of lamination called an essential lamination. These need not be measured, but are “end-incompressible”, which means, roughly speaking, that the lamination has no infinitely long folds. They develop a concept of essential branched 2-manifold, proving that all the laminations (fully) carried by an essential branched 2-manifold are essential and that every essential lamination is carried by some essential branched 2-manifold. Part of the definition of an essential branched 2-manifold is the requirement that it carries at least one lamination; the authors show that every 3-manifold contains a branched surface satisfying every other requirement to be essential, except possibly that one. They display examples showing that “essential laminations occur quite frequently in non-Haken manifolds”, although it is unknown exactly which 3-manifolds contain them.

These are important and difficult technical results, but the showpiece of the paper is he following generalization of Waldhausen’s theorem: if a closed 3-manifold contains an essential lamination, then its universal cover is 3-dimensional Euclidean space (the obvious necessary conditions of irreducibility and asphericity are built into the definition of essential lamination). Waldhausen’s original proof involved showing that the universal cover has the structure of a tree whose edges are the preimages of the surface, and whose vertices are the components of the preimage; inductively, these components have Euclidean interiors. In the lamination version, one uses the essential branched 2-manifold in place of the incompressible 2-manifold, but it turns out that the corresponding graph need not be a tree; it can contain loops. To reduce to a compact problem, the authors make use of the fact that the universal cover is Euclidean if every finite 1-complex imbedded in it lies in a 3-cell. A delicate argument shows that by “splitting” the branched surface, the portion of the graph that meets a given 1-complex can be changed into a tree, from which it follows that the complex lies in a 3-cell.

Non-Haken 3-manifolds (other than those whose universal cover is the 3- sphere), are still rather poorly understood. By the mid 70’s, there were only a few examples known of non-Haken 3-manifolds, and it was felt that they were probably quite rare. This feeling turned out to be totally incorrect, because Thurston showed how to construct huge quantities of non-Haken 3-manifolds. Thurston’s method shows that “most” ways of attaching solid tori to the torus boundaries of hyperbolic knot and link complements result in non-Haken 3-manifolds.

Currently there is much effort to try to understand non-Haken 3-manifolds by using laminations to play the role that 2-dimensional submanifolds do in the Haken theory. A lamination is a foliation of a closed subset of a manifold, whose leaves have codimension 1 in the manifold. The systematic use of laminations in low-dimensional topology was initiated by Thurston, who introduced the device of “train tracks”, or branched 1-dimensional submanifolds of 2-manifolds. By assigning numbers, called weights, to the tracks in a train track, one can construct the laminations “carried” by the train track. In this way, a train track parametrized a portion of the space of (measured) laminations in the 2-manifold. Laminations in 3- manifolds are studied using the 2-dimensional analogue of train tracks: branched 2-manifolds. Through work of Floyd, Gabai, Hatcher, Morgan, Oertel, Shalen, and others, considerable technical machinery has been developed for working with laminations by using branched 2-manifolds.

In this paper, the authors give two equivalent definitions of a kind of lamination called an essential lamination. These need not be measured, but are “end-incompressible”, which means, roughly speaking, that the lamination has no infinitely long folds. They develop a concept of essential branched 2-manifold, proving that all the laminations (fully) carried by an essential branched 2-manifold are essential and that every essential lamination is carried by some essential branched 2-manifold. Part of the definition of an essential branched 2-manifold is the requirement that it carries at least one lamination; the authors show that every 3-manifold contains a branched surface satisfying every other requirement to be essential, except possibly that one. They display examples showing that “essential laminations occur quite frequently in non-Haken manifolds”, although it is unknown exactly which 3-manifolds contain them.

These are important and difficult technical results, but the showpiece of the paper is he following generalization of Waldhausen’s theorem: if a closed 3-manifold contains an essential lamination, then its universal cover is 3-dimensional Euclidean space (the obvious necessary conditions of irreducibility and asphericity are built into the definition of essential lamination). Waldhausen’s original proof involved showing that the universal cover has the structure of a tree whose edges are the preimages of the surface, and whose vertices are the components of the preimage; inductively, these components have Euclidean interiors. In the lamination version, one uses the essential branched 2-manifold in place of the incompressible 2-manifold, but it turns out that the corresponding graph need not be a tree; it can contain loops. To reduce to a compact problem, the authors make use of the fact that the universal cover is Euclidean if every finite 1-complex imbedded in it lies in a 3-cell. A delicate argument shows that by “splitting” the branched surface, the portion of the graph that meets a given 1-complex can be changed into a tree, from which it follows that the complex lies in a 3-cell.

Reviewer: D.McCullough

##### MSC:

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

57R30 | Foliations in differential topology; geometric theory |