By Matthew G. Brin

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Let M be a compact, connected submanifold of V so that V is end irreducible rel M. 2. 1. 5. Let V be a connected, orientable, eventually end irreducible, end 1-movable 3-manifold. Let M be a compact, connected submanifold of V so that V is end irreducible rel M. 2. 1. 5. 2]. This theorem has some exclusions that do not apply here since the theorem will only be invoked for irreducible, orientable 3-manifolds. 2: Note that if L is a compact, connected submanifold of V that contains K, then V is also end irreducible rel L.

Each /,• is a proper map from a subset of D2 that contains (D2 — Y) into U. Its properties will be made clear by defining /;+i as a modification of /,•. Let E\ — /tr"1(Ar«) • Let E\ be the union of E'{ and all components of D2 — E\ that lie in the domain of /,-. The surface E1- is compact and some Mj contains fi(E*{). By taking a subsequence and renumbering, we may assume that fi(E*{) C M,-+i. If fi(E*) does not lie in iV»+i, then / r 1 ( F r JVt-+i) HE* consists of simple closed curves in the interiors of the components of D2 — E[ that lie in E\ .

We will refer to the union of the surfaces D(H) n U — iV(i, j — i), where the union is over all 2-handles H of T(i), as the compression tracks of T(i) outside iV(i, j — i). We will use QR to denote the procedure described above that takes iV(i, j-f 1 — i) to N(i —l,j — i). The handles of QR are contained in (J7 — Mi). Since Fr N(i, j —i) is incompressible in {/ — M«, we can assume that the handles in QR miss N(i,j — i). 8, the compression tracks of T(i) outside iV(i, j — i) are incompressible in U — N(i, j — i).