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By Matthew G. Brin

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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. 2 the handle procedure PQ need not be a compression procedure for (M, N). 2. We recall that item (N2) in the definition of normality requires that Pj D Pi have the form D(Pi) x Ji(j) in Pi where Pj and Pt- are a 2-handle and a 1-handle respectively with j > i in a normal handle procedure P. For each 1-handle P,- in a normal handle procedure P , let Jt- be the union of the intervals J t (j) as in the previous sentence as j ranges over those integers from (z + 1 ) through the length of P for which Pj is a 2-handle.

Ni(A'i+1 — A\) (since the frontier of A[ in A'i+1 is Fi), and thus (A'i+1 — A'{) is also homeomorphic to Fi+i x [0,1], We pick a homeomorphism that takes the pair (-A(-+1 — A'if Fj+i) to the pair ( F , + 1 x [0,1] , (Fi+1 x {1}) U (dFi+i x [0,1])). This homeomorphism carries Fi into F, + i x {0} and it gives us a specific embedding hi of Fi into F t - + i. -+1 x {i + 1}) U (dFi+1 x [0, i + 1]) , (FiXii})\J(dFix[0,i\)). With the above structure for the sets A\, the union of the A\ has the form R x [0,oo) where the surface R is the direct limit of the Fi under the above embeddings.

Let P be a normal handle procedure for (M, N). We will say that P is taut if the following two conditions hold. ) ( T l ) For each 2-handle Pj in P , the set Pj C\FTN viewed as a subset of Pj has the form [D(Pj) n FriV] x 7. (T2) For each 2-handle Pj in P there exists no other 2-handle Pj for the pair (MiNiPi-1)) so that (i) dD(Pj) = dD(Pj)1 (ii) Pi~lPj is a normal handle procedure that also satisfies ( T l ) above, and (iii) the number of components of D(Pj) n N is less than the number of components of D(Pj) n N.

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