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By Matthew G. Brin

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Now choose the corresponding subsequence of (Ni). 3 are easy exercises in cutting objects off on the core disks of the 1-handles. We will use the two lemmas that follow to prove item (v), the inheritance of end 1-movability. The first of the two lemmas is more general than needed, but may be of interest in its own right. We first need a definition. If (D2 —Y) is a virtual disk, then we refer to (D2 — Z) as a partial compactification of (D2 — Y) if (D2 — Z) is a virtual disk and there is a closed subset Y' of Y that maps continuously onto Z.

Then U — G is end 1-movable. P R O O F : The outline of the proof consists mainly of a sequence of reductions. In each step we will be working with an orientable, end 1-movable 3-manifold, but the setting will change from step to step. Let W — U — G. The main theorem discusses W. The first step reduces the main theorem to a statement about U. Next we reduce this to a statement about end reductions of U. Next comes a statement about manifolds where irreducibility and eventual end irreducibility are assumed, and lastly we work in a setting in which one endedness is also assumed.

THICKSTUN 28 for a small open collar on dFiMj^ in FrM,*.. , then P will be disjoint from (FiMjj — FTMJ^) for i < k, and thus P can also be regarded as a handle procedure for (U,Mjyi) for any i < k. We are now ready to describe how Mj is altered to create exhaustion elements for the various end reductions VJ of (£/, Pi, (M,-+fc)) at Mt-. Note that if j < i, then there is no alteration. Thus, for each i < j , we will describe how to alter Mj to give an exhaustion element N(i, j — i) of VJ. ) Two points of view will be maintained simultaneously during this process.

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