By Matthew G. Brin

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Items (iii) and (iv) can both be accomplished since it is assumed about the exhaustion element, N(i—l}j — i), of Vf_i that it has been proviously constructed (from M j _ i ), that it is contained in JV(i, j—i), and that it has Fr N(i— 1, j—i) incompressible in (U—Mi-i). Item(v) will guarantee that P(j, j—i)Q(j, j—(i—l)) is a valid handle procedure for Mj^ f° r a n v k J ~ W - J - (*' - 1))] - Gi-U . and we use N(i, j — i) to denote the set defined by either side.

We let Mjti = Mj - Gi}j. Thus Mjti is the exhaustion element Mj "floated" away from the surface Gt-. Note that the "distance" that Mj moves away from Gi in the collar on dU to create Mjj is independent of i, but that Mj is pulled farther in than Mk if j < k. Since (Mj DdU) is disjoint from both G,- and the collar on dGi in Fi whenever j < i, we have Mj%i = Mj if j < i. We have Mj}i C Mj^+i and Mjyi C IntMj +1) ,for each i and j . If t < i , then FvMj}i D FrMj k is all of FrM^* except MATTHEW G.

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. We also note that dropping a given number of rightmost entries from an n-tuple is an order preserving operation for the order that we have put on tuples. Thus facts about the spaces N(i, j) and the procedures that changes the spaces N(i, j) into the spaces N(i — k, j + k) will follow when we prove those facts in the special case where j = 0.

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