Essential Laminations in Seifert- bered Spaces

Over the past few decades, the importance of the incompressible surface in the study of 3-manifold topology has become apparent. In fact, nearly all of the important outstanding conjectures in the eld have been proved, for 3-manifolds containing incompressible surfaces (see, e.g., [20],[22]). Faced with such success, it becomes important to know just what 3-manifolds could contain an incompressible surface. Historically, the rst 3-manifolds (with in nite fundamental group) which were shown to contain no incompressible surfaces were a certain collection of Seifertbered spaces. Waldhausen [21], in the 1960's, showed that an incompressible surface in a Seifertbered space is isotopic to one which is either vertical or horizontal. This added structure puts a severe restriction on the existence of an incompressible surface, and led to the discovery of these `small' Seifertbered spaces. Now in recent years the essential lamination, a recently-de ned hybrid of the incompressible surface and the codimension-one foliation without Reeb components, has begun to show similar power in tackling problems in 3-manifold topology (see [7]). It also has the added advantage of being (seemingly) far more widespread than either of its `parents'; its more general nature makes it far easier to construct in a wide variety of 3-manifolds (see, e.g., [6]). In light of these facts, it would be interesting to know if there are any 3-manifolds which contain no essential laminations, and only natural to look in the same place that Waldhausen found his examples.