When Incompressible Tori Meet Essential Laminations

Roussarie and Thurston [Ro], [Th] showed that given a taut foliation F and an incompressible torus T in a 3-manifold M , the torus may be isotoped so that it is either transverse to the leaves of the foliation, or is equal to one of the leaves of the foliation. This result was known to be false in the more general context of a foliation F without Reeb components. The only obstruction, however [Ro], is the existence of a ‘cylindrical component’ of the foliation. This is an I-bundle over a torus or Klein bottle, which is saturated by the leaves of the foliation. The boundary component(s) of the bundle are leaves, and the interior of the bundle is foliated by open annuli and Möbius bands, which spiral in towards the boundary components in the same direction (see, e.g., [Ro, p. 109]). Even when F is taut and transverse to T , the foliation F|T (see §1 for notation) need not be taut; it may have half-Reeb components, which are solid tori in M |T bounded by an annulus leaf of F|T and foliated as ‘half’ of a Reeb component (see Figure 1). However, if the foliation is C(2), an argument due to Hass shows that such components can be removed by a further C(2) isotopy of F (see [BNR]). In this paper we show that these facts can be extended to C(0) foliations, and in fact to essential laminations: Theorem. Given an essential lamination L and an incompressible torus T in a 3-manifold M , then, possibly after splitting L open along a finite number of leaves, either L can be isotoped so that it is transverse to T and L|T is essential in M |T , or L has a cylindrical component.