# 3 Manifolds Which Are End 1 Movable by Matthew G. Brin PDF

By Matthew G. Brin

ISBN-10: 0821824740

ISBN-13: 9780821824740

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**Extra info for 3 Manifolds Which Are End 1 Movable**

**Sample text**

An) is less than an (n -f 1)-tuple (60, • • • , bn) if the smallest value of k with ak ^ bk has ak < bk. Tuples with an unequal number of entries will not be compared. Note that we do not consider intersections of the core disks with the sets N(—> —). We now put restrictions on the selections of the 2-handles of the procedures Q and R. Let S be either the procedure Q(j) j — (* ~ 1)) o r the procedure R(j, j — (i — 1)) consisting of the sequence of handles (Si). Let T be the submanifold for which 5 is a handle procedure.

Since V is an open subset of U, and since F is compact and properly embedded in V, it is properly embedded in U. 4 that F is incompressible in U. 2, and it is incompressible in U because it is simply connected. Now let B be as in the statement of the lemma. Let W be the component of U — K - i that contains F. We will show (1) that if Ko n W contains a loop that is non-trivial in U, then wiF —• niB is onto. Then we will show (2) that if A'o H W does not contain a loop that is non-trivial in U, and if wiF —» T^IB is not onto, then there is a compact, connected set KQ in U that contains KQ and MATTHEW G.

However each M( is compact, so a subsequence of (M() may be chosen so that M{ C Int M / + 1 for each i. 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.

### 3 Manifolds Which Are End 1 Movable by Matthew G. Brin

by Anthony

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