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By Matthew G. Brin

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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. BRIN AND T. L. THICKSTUN 28 for a small open collar on dFiMj^ in FrM,*..

Further assumptions about the procedures will become apparent from the requirements that we will put on the procedures for constructing the next string of spaces. The "ultimate" procedure that operates on N(j, 0) is P(j, j) which is a handle procedure for Mj,o. We choose Mj+i to contain N(j, 0) U P ( j , j) by dropping a finite number 3-MANIFOLDS WHICH ARE END 1-MOVABLE 31 of terms from the exhaustion (Mk) of U. Again we start with N(j -f 1, 0) = Mj+i and with the empty procedure P(j-\1,0). We put no requirements on P( j - f 1,0).

4. Let U be an end 1-movable 3-manifold. There is a compact, connected set KQ in U so that if either (i) V is a reduction of U with KQ C V, F is a compact, connected, properly embedded surface in a component V of V with V Pi Ko = 0, and F is incompressible in V, or (ii) F is a compact, connected, properly embedded surface in U that is simply connected and that lies in U — Ko, then F separates U, F is incompressible in U, and ir\F —• niB is onto, where B is the closure of the component of U — F that is disjoint from Ko .

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