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**Essential laminations in Seifert-fibered spaces.**
*(English)*
Zbl 0791.57013

The fundamental work of Waldhausen in 1968 provided a condition – the existence of a (compact) 2-sided incompressible surface not a 2-sphere – which enables strong results to be proved for compact (orientable) irreducible 3-manifolds. For example, the universal cover is homeomorphic to a subset of the closed 3-ball which contains the interior. Additionally, much information about the homotopy equivalences and homeomorphisms of such manifolds can be obtained. It appeared that almost all irreducible 3-manifolds, except for those with finite fundamental group and a few special examples of Seifert fibered spaces noticed by W. Heil, contained the needed incompressible surface. However, in the late 1970’s W. Thurston proved that this view was quite incorrect, by showing that vast numbers of examples of closed hyperbolic 3-manifolds containing no incompressible surface could be constructed by attaching solid tori to link complements.

Finding substitutes for the incompressible surface has been difficult, but some success was achieved by the use of laminations, which are closed subsets having local neighborhoods \([0,1]\times D^ 2\) in which they appear as \(C \times D^ 2\) for some closed subset \(C \subseteq [0,1]\). Laminations decompose into surfaces, called the leaves, which may be compact but often are noncompact. Certain technical incompressibility conditions are needed to get results, and laminations satisfying these are called essential. A striking success of this approach was the work of D. Gabai and U. Oertel [Essential laminations in 3-manifolds, Ann. Math., II. Ser. 130, No. 1, 41-73 (1989; Zbl 0685.57007)] in which the authors proved that the universal cover of an irreducible 3-manifold containing an essential lamination is a subset of the 3-ball containing the interior. No examples were known of irreducible 3-manifolds with infinite fundamental group which did not contain essential laminations, so conceivably these might provide the technical means for the full extension of Waldhausen’s results.

In this paper, the author examines essential laminations in Seifert- fibered spaces, in particular showing that the Heil examples do not contain essential laminations. The basic idea is to prove an analogue for laminations of the “vertical-horizontal” theory, which asserts that an incompressible surface in a Seifert-fibered space is isotopic either to one which is a union of fibers (vertical) or to one which meets the fibers transversely (horizontal). The main theorem is that every essential lamination contains a sublamination which is isotopic to either a vertical or horizontal lamination. Some of the Heil examples do not contain a vertical or horizontal essential lamination, and hence no essential laminations at all. As another application, the author shows that if an essential lamination with no compact leaves in a closed orientable Seifert-fibered 3-manifold contains a horizontal sublamination, then the lamination is isotopic to a horizontal one. Another application involves essential foliations in Seifert-fibered spaces.

The proof of the main theorem is long and elaborate. The basic setup is to regard the Seifert-fibered space as a union of fibered solid tori (for example, the preimages of the 2-simplices of a triangulation of the orbit surface, fine enough so that no 2-simplex contains the image of more than one exceptional orbit). A very careful examination of the intersection of the lamination with a single one of these solid tori is made, and it is shown that the lamination can be moved by isotopy into good position with respect to the solid tori. Eventually the intersections can be pieced together to prove the theorem.

Finding substitutes for the incompressible surface has been difficult, but some success was achieved by the use of laminations, which are closed subsets having local neighborhoods \([0,1]\times D^ 2\) in which they appear as \(C \times D^ 2\) for some closed subset \(C \subseteq [0,1]\). Laminations decompose into surfaces, called the leaves, which may be compact but often are noncompact. Certain technical incompressibility conditions are needed to get results, and laminations satisfying these are called essential. A striking success of this approach was the work of D. Gabai and U. Oertel [Essential laminations in 3-manifolds, Ann. Math., II. Ser. 130, No. 1, 41-73 (1989; Zbl 0685.57007)] in which the authors proved that the universal cover of an irreducible 3-manifold containing an essential lamination is a subset of the 3-ball containing the interior. No examples were known of irreducible 3-manifolds with infinite fundamental group which did not contain essential laminations, so conceivably these might provide the technical means for the full extension of Waldhausen’s results.

In this paper, the author examines essential laminations in Seifert- fibered spaces, in particular showing that the Heil examples do not contain essential laminations. The basic idea is to prove an analogue for laminations of the “vertical-horizontal” theory, which asserts that an incompressible surface in a Seifert-fibered space is isotopic either to one which is a union of fibers (vertical) or to one which meets the fibers transversely (horizontal). The main theorem is that every essential lamination contains a sublamination which is isotopic to either a vertical or horizontal lamination. Some of the Heil examples do not contain a vertical or horizontal essential lamination, and hence no essential laminations at all. As another application, the author shows that if an essential lamination with no compact leaves in a closed orientable Seifert-fibered 3-manifold contains a horizontal sublamination, then the lamination is isotopic to a horizontal one. Another application involves essential foliations in Seifert-fibered spaces.

The proof of the main theorem is long and elaborate. The basic setup is to regard the Seifert-fibered space as a union of fibered solid tori (for example, the preimages of the 2-simplices of a triangulation of the orbit surface, fine enough so that no 2-simplex contains the image of more than one exceptional orbit). A very careful examination of the intersection of the lamination with a single one of these solid tori is made, and it is shown that the lamination can be moved by isotopy into good position with respect to the solid tori. Eventually the intersections can be pieced together to prove the theorem.

Reviewer: D.McCullough (Norman)

### MSC:

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

57R30 | Foliations in differential topology; geometric theory |