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**Example text**

Our K is K-i U KQ . Let M be a compact, connected submanifold of V that contains K. It is trivial that V is end irreducible rel M. Let (N t ), i > 0, be an exhaustion of V so that each TV,- is connected, so that M C Int. JVi, and so that each Fr JV,is incompressible in (V — M). Let No denote M . 2, we know that for each i > 0, each component of V — Ni has connected frontier. PROOF: 36 3-MANIFOLDS WHICH ARE END 1-MOVABLE 37 For each i > 0, Ni+i — Ni is compact. Thus there is a finite collection {C,j} of pair wise disjoint compact 3-manifolds in iVi+i — Ni so that the frontier of each C{j in Ni+i — Ni is a 2-sphere in (iVj+i — N{), and so that if Li is formed from Ni+i — N( by replacing each C,-j by a 3-cell, then Li is irreducible.

Each hi induces an injection on TTI . Thus the natural embedding of each Fi into R induces an injection on TTI . If all but finitely many hi induce surjections on 7Ti, then it is easy to show that the hypothesis that iriFi —* niBi is not onto would fail for all sufficiently large i. Now if R were of finite type, then any representation of R as an ascending union of compact surfaces Fi would have to have some Fj containing a generating set of loops for itiR. If every TTiFi —* 7TiR is one to one, then i > j would imply that T\Fi —• TTIR and ^l-Ft'+i - • KiR would be isomorphisms and thus TT\Fi —+ iriFi+i would be an isomorphism.

T2) For each 2-handle Pj in P there exists no other 2-handle Pj for the pair (MiNiPi-1)) so that (i) dD(Pj) = dD(Pj)1 (ii) Pi~lPj is a normal handle procedure that also satisfies ( T l ) above, and (iii) the number of components of D(Pj) n N is less than the number of components of D(Pj) n N. The notion of tautness will be redefined below for certain sequences of handle procedures. If P is a normal handle procedure for (M, iV), then the the union of 3-MANIFOLDS WHICH ARE END 1-MOVABLE 27 the sets D(Pj) — N, where the union is over all the 2-handles Pj in P , will be called the compression track T(P) of P.