By Matthew G. Brin

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4. The argument that replaces W by an irreducible companion of W is almost identical. 5 by using the hypothesis that TT\FIW —• Tr\W is an isomorphism. 5 by using both the fact that each 2-sphere is contained in some A,- and the hypothesis that ir\Fi —• 7TiA,- is an isomorphism. We now study the structure of W assuming that W is irreducible. Let (M,), i > 0, be a connected exhaustion of V with Mo = M so that each FrM» is incompressible in V — M. We may assume that each complementary domain of Mi in V is unbounded.

L. O) C iV(2,0) C iV(3,0) C - in y in • in s in • JV(0,1) C JV(l, 1) C iV(2,l) C JV(3,1) C - in / N(0,2) in s in / in s C W(l,2) C AT(2,2) C iV(3,2) C - in y in / in ^ in Figure 1 /,-+i be an embedding which can be accomplished by eliminating some curves of f~1(FiNi+i) that might not lie in E*. This may cause fi+i to have a larger domain than /,-. The alterations are all to be done so that the disks in D2 on which fi is altered are mapped by fi+i very near FrA^+i. This insures that the image of / l + i is disjoint from K (assuming that this is true for fi).

In addition, if Fr M is connected, then F separates N. P R O O F : If the first conclusion is false, then a simple closed curve J in M would exist that pierces F exactly once. But then no homotopy would pull J off F. If the second conclusion is false, then a simple closed curve J in N would exist that pierces F exactly once. But the connectivity of F r M would allow J to be replaced by a simple closed curve in M that pierces F exactly once. 2. Let U be a connected, end 1-movable 3-manifold and let K be a compact submanifold of U so that loops in U — K push to the ends of U in U, and so that each complementary domain of K in U has connected frontier.