By Matthew G. Brin

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This is done precisely in the description of a type II end reduction in Section 1. We assume that Go is contained in each element of (Mi). We must find for each Ni an Nj so that loops in W — Nj push to the ends of W in W — Ni. It is sufficient to find an Nj so that for any k > j , loops in W — Nj homotop in W — N{ to loops in W — Nk. 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.

In addition: (i) If an initial segment R of P is normal, then Q can be chosen so that R is an initial segment of Q. (ii) IfFvN(P) is incompressible in M, then so is FTN(Q). (iii) If P is a compression procedure, then Q will also be a compression procedure, (iv) If O is an open set in M containing P, then Q can be chosen so that O also contains Q. 2. If P is a normal handle procedure for (M,N) with N compact, then there is a compression procedure Q for (M, N{P)) so that PQ is normal and so that FiN(PQ) is incompressible in M.

5 (Nesting). The proof is conceptually simple, although the notation is complicated by the need to keep track of several interlocking sequences of objects and events. We first discuss the idea of the proof. Our discussion will refer to Figure 1 where each column shows an end reduction VJ- of U at Mi and the exhaustion elements N(i,j) of Vi. An arrow from N(i, j) to N(i— 1, j-\-1) represents the fact that N(i— 1, j - f 1) is to be constructed by modifying N(i,j). The modification will be accomplished by combining compressions with the operation of "floating away" from part of the boundary.