By Matthew G. Brin
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Those notes have been the foundation for a sequence of ten lectures given in January 1984 at Polytechnic Institute of latest York below the sponsorship of the convention Board of the Mathematical Sciences and the nationwide technology starting place. The lectures have been aimed toward mathematicians who knew both a few differential geometry or partial differential equations, even supposing others might comprehend the lectures.
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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.