By Matthew G. Brin
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5. 3 applies to each Vi. 3. Let K\ be a compact subset of U. Each Mt- contains Ko, and some Mj contains K\. 3 to the end reduction Vj of U at Mj . 3. Let K$ be a compact subset of U that contains K2. Choose a VJ.
2(11). Note that Ai contains W Ci Mi. 1, we know that each component of F r M; other than Fi separates A,*. Let G be a component in W of FrM, other than F,-. Let C be the component of A,- — G that misses M,-. We have that Fr C = G. The set C is a complementary domain of M,- D W in A,-, and every such complementary domain arises in this way. The set C is also a complementary domain of Mt- 3-MANIFOLDS WHICH ARE END 1-MOVABLE 41 in V, and every such complementary domain, except the one whose frontier is F , , arises in this way.
Consider any j > i large enough so that loops in U — Mj push to the ends of U in U — Mi. This implies that any loop in (17 — Mj) n W pushes to the ends of W in W — Ni. If A is a loop in W — Nj that does not lie in U — Mj , then (assuming general position) A is composed of a finite number of paths, each with endpoints in Fr Mj , with one set of the paths lying in U — Mj , and with the other set lying in a collar on G n Mj . In order to push A into W — Nk , it suffices to push the subpaths of A that lie in U — Mj into U — Mk plus a collar on G.
3 Manifolds Which Are End 1 Movable by Matthew G. Brin