Essential laminations and deformations of homotopy equivalences, II: the structure of pullbacks

It is a long-standing conjecture in 3-manifold theory that every homotopy equivalence f :M!N between closed, irreducible 3-manifolds M and N , with j 1(M)j (=j 1(N)j) = 1, is homotopic to a homeomorphism. In [Wa], Waldhausen proved that this conjecture was true, if we assume that N contains a 2-sided incompressible surface S. The proof consists of rst homotoping the map f so that the pullback surface f (S) is an incompressible surface, and then splitting both manifolds open along these surfaces, and proceeding by induction on a hierarchy of N a collection of incompressible surfaces in the successively split-open manifolds which end up splitting N into a 3-ball. The base case is the Alexander trick. In [Br1], we began a program to extend this theorem to the case that N is laminar, i.e., N contains an essential lamination [G-O]. We showed that, given a homotopy equivalence f :M!N between non-Haken, (irreducible) 3-manifolds and a transversely-orientable essential lamination L N , if the pullback lamination f (L) M is essential then we could homotope the map f to a homeomorphism. In this paper we study the `other half' of Waldhausen's proof when can one deform a homotopy equivalence to give an essential pullback by trying to understand what structure the pullback of an essential lamination has, in general, under a homotopy equivalence. We work, as in [Br1], under the hypothesis that M and N are nonHaken; otherwise, all that we shall use about the map f is that it has degree 1 (in particular, non-zero degree). It has been conjectured that the method employed by Waldhausen surgeries of the pullback achieved as homotopies of the map f will be successful (with some modi cation) in creating an essential pullback in the present context. We begin this paper by showing that if the homotopy-through-surgery process will succeed in making the pullback essential, then, at the very beginning, the pullback must have had a fairly restrictive structure each leaf of the pullback must map to its corresponding leaf in L with degree 0 or 1. In the remainder of the paper we attempt to show that leaves of a pullback do in fact have this property. We must admit at the outset that we do not succeed in proving this but we will see that the