Incompressible Maps of Surfaces: Boundary Slopes and Dehn Filling

The best understood 3-manifolds are Haken 3-manifolds; these are manifolds that are irreducible and contain incompressible surfaces. One approach to 3-manifolds is to try to generalize the notion of an incompressible surface. Essential laminations and mapped-in π1-injective surfaces have both been used with some success as substitutes for incompressible surfaces, yielding theorems which apply to classes of manifolds much larger than the class of Haken manifolds. In dealing with maps of surfaces f : S → M which induce injections on π1, a proof of the Simple Loop Conjecture would be helpful. Suppose M is a compact, orientable closed 3-manifold, S is a closed orientable surface of genus g ≥ 1, and f : S → M is a map with the property that the induced map π1(S) → π1(M) has non-trivial kernel. Then the conjecture is that there is an essential simple closed curve γ in S, with f |γ null-homotopic in M . If the conjecture is true, a map f : S → M which is not essential in the sense that it induces an injection on π1 can be replaced by a map of a surface of lower genus by performing surgery on S and on the map f . In dealing with embedded incompressible surfaces, this kind of surgery is an important tool. For completeness, we give a precise definition of injective and ∂-injective maps of surfaces. We always assume M is an orientable, compact 3-manifold. If S is a compact surface, possibly with boundary, we will say that a map f : (S, ∂S) → (M,∂M) is π1-injective if the induced map π1(S) → π1(M) is injective. We will say the map f : (S, ∂S) → (M,∂M) is ∂-π1-injective if the induced map on π1(S, ∂S, p) → π1(M,∂M, f(p)) is injective for every choice of base point p in ∂S. One can define a class of maps of surfaces which would be π1-injective if the Simple Loop Conjecture were true. Given an orientable surface S which is not a sphere or disc, we shall say that a map f : (S, ∂S) → (M,∂M) is incompressible if no essential simple loop in S is mapped to a homotopically trivial curve in M . When M has boundary, we say that the map f : (S, ∂S) → (M,∂M) is ∂-incompressible if no essential simple arc in S is mapped to an arc in M which is homotopic in M to an arc in ∂M . If f is a sphere (or a disc) a map f : (S, ∂S) → (M,∂M) is essential if it is not null-homotopic (not homotopic to a disc in ∂M). We will say a