Essential laminations in Seifert-fibered spaces: Boundary behavior
This paper continues the author's work on essential laminations in Seifert-fibered spaces, Topology 32, No. 1, 61-85 (1993; Zbl 0791.57013)]. There it was shown that every essential lamination contains a sublamination for which every leaf is either a union of the circle fibers (vertical) or meets the fibers transversely (horizontal). This paper extends that result to Seifert-fibered spaces, \(M\), with non-empty boundary. If the lamination, \(L\), contains a vertical sublamination then it can be isotoped so that each leaf is either vertical or horizontal. The situation is more complicated if \(L\) contains a horizontal sublamination. In [Essential laminations in \(I\)-bundles, Trans. Am. Math. Soc. 349, No. 4, 1463-1485 (1997)] the author showed, in the case when \(M\) is closed, that \(L\) can be isotoped so that each leaf is eiter horizontal or vertical except for the leaves in `Reeb sublaminations'. This paper extends the definition of a Reeb sublamination to apply in the case that \(M\) has a nonempty boundary. In the case that \(L\) contains a horizontal sublamination it is possible that the lamination can contain two parallel horizontal annuli with Reeb-type leaves lying in between; these leaves spiral toward the bounding annuli. Such a leaf cannot be isotoped to be vertical or horizontal in \(M\) and, in this work, it is shown that such leaves are essentially the only counterexamples. Since there are only a few, known, orientable Seifert-fibered spaces which contain horizontal annuli this case is rare.
scientific article; zbMATH DE number 1317240
scientific article