## 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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**Sample text**

In addition, if Fr M is connected, then F separates N. P R O O F : If the first conclusion is false, then a simple closed curve J in M would exist that pierces F exactly once. But then no homotopy would pull J off F. If the second conclusion is false, then a simple closed curve J in N would exist that pierces F exactly once. But the connectivity of F r M would allow J to be replaced by a simple closed curve in M that pierces F exactly once. 2. Let U be a connected, end 1-movable 3-manifold and let K be a compact submanifold of U so that loops in U — K push to the ends of U in U, and so that each complementary domain of K in U has connected frontier.

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). We now assume that N(i, j +1 — i) has been constructed with a procedure P(j + l , j 4- 1 — 0- We consider the "ultimate" procedure T(i) = P(j,j) — P(j,j — i) which operates on N(i,j — i) to produce N(0,j). The difference makes sense because P(j,j — i) is an initial segment of P(j, j). We assume that N(i, j—i)lfT(i) is contained in N(i, j+l—i). We will refer to the union of the surfaces D(H) n U — iV(i, j — i), where the union is over all 2-handles H of T(i), as the compression tracks of T(i) outside iV(i, j — i).

II. There is a homeomorphism from W to a vaguely punctured (R x [0,00)) — L where R is an open, orientable, connected surface having infinitely generated first homology, where L is a closed subset of R x {0} so that the components of L are compact, and where the homeomorphism carries FiW to a compact submanifold of Rx {0} that is disjoint from L. 2. Let V be a connected, orientable, eventually end irreducible, end 1-movable 3-manifold, and let K be a compact subset of V so that V is end irreducible rel K.

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

by Edward

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