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For each CQJ , we let SOJ be the 2-sphere that is the frontier of CQJ in N\ — No. We let CQJ be the closure of the component of V — SQJ that contains CQJ . We note that no CQJ contains Fr No . We form VQ by replacing each CQJ by a 3-cell. Each element of N{, for i > 0, is altered by having each Ni D CQJ replaced by a 3-cell. We do not invent a new notation for what results. We do note that the new version of Ni contains all the 3-cells that were introduced in the above operation, and that VQ minus the new N\ is a subset of V minus the old JVi that is disjoint from all of the CQJ.

This gives another restriction on how we choose handles in our compression procedures. The complications of this point of view arise from the fact that X will go through further compressions at a later time to help build a smaller end reduction. The handles that are used to do this further compression on X must also obey certain restrictions so that they can be successfully avoided as well. To keep incompressible those portions of the cores of P that lie outside of X, we make use of the techniques of [BT1] based on objects known as compression tracks.

To make the rest of the description easier, we start with N(j, 0) = Mjj = Mj , and P ( j , 0) = 0. We observe that P ( j , 0) is a valid handle procedure for Mj,k for any k < j . We now assume that N(i,j — i) has been created by applying a normal handle procedure P(j,j — i) to Mj,,-, and that P(j,j — i) has been chosen so that it is also a valid handle procedure for Mjyh for a n v k < i. We create N(i — \>j — (i — 1)) in two steps from N(i,j — i). ) First we build a normal compression procedure Q(j,j — (i — 1)) for N(i,j — i) so that (i) QtJJ-ii-l^ciU-Mi^), (ii) P ( j , j - i)Q(j, j - (i - 1)) is normal, (iii) FrN(i, j — i)Q(j, j — (i — 1)) is incompressible in (U — M«_i), (iv) <2(j, j — (i — 1)) is disjoint from N(i — 1, j — i), and ( v ) Q(i) J - (z* - 1)) i s disjoint from (dU x [0, a ; ]) n Int Mj}i.

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