By Matthew G. Brin

ISBN-10: 0821824740

ISBN-13: 9780821824740

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This will guarantee that JV(i, j - i) U T(i) is contained in N(i, j + 1 - i)(QR) = N(i - 1, j - i), and as a result that N(i — 1, j — (i — 1)) U T(i — 1) is contained in N(i — 1, j — i). Thus the inductive assumptions are maintained. This gives us the containments that we want and completes the proof. | 2. ELEMENTARY CONSEQUENCES OF END 1-MOVABILITY The main technique of the paper is to analyse end 1-movable 3-manifolds by using end reductions to reduce the analysis to end 1-movable, eventually end irreducible 3-manifolds.

3 ( V ) : 3-MANIFOLDS WHICH ARE END 1-MOVABLE A(P) A C s in s C *(Q) 25 B Figure 2 and so that any loop in (17 — Mk) pushes to the ends of U in (U — Mj). Let a be a loop in (V — Nk). 6, a is homotopic in (U — Nk) to a loop a in (J7 — M&). Now a pushes to the ends of U in (U — Mj). Since we get JVj from Mj by cutting 1-handles, we have Nj C Mj . Combining two homotopies, we have that a pushes to the ends of U in (U — Nj). The half open annulus that is the domain of the combined homotopy is a virtual disk (D2 — {x}) that has exactly one end.

T3) For each 2-handle 5/ in S and for each k with 0 < k < (j — i), the set 5/nFrJV(t + J b , i - ( i + fc)) viewed as a subset of Sk has the form [D(Si) n YvN(i + Jb, j - (i + k))] x / . (T4) For each 2-handle 5/ in S there is no 2-handle S( for (t^rfS'"" 1 )) that satisfies (i) dD(S',) = dD(S,), 30 MATTHEW G. BRIN AND T. L. THICKSTUN (ii) Sl 1S{ is a normal compression procedure for (Uy T) that also satisfies item (T3) above, and (iii) 7(5/) is less than 7(5/)S. REMARK: We note that the process for obtaining any N{i — k,j -f k) from N(i,j) is identical in nature to the process for obtaining any N(p— q,q) from N(p, 0) = Mp.

### 3 Manifolds Which Are End 1 Movable by Matthew G. Brin

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