## Read e-book online 3 Manifolds Which Are End 1 Movable PDF

By Matthew G. Brin

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Let V be as in the statement of the lemma. Let V and F be as described in item (i). 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.

We assume that all procedures involved satisfy items (T3) and (T4) above. 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 recall that item (N2) in the definition of normality requires that Pj D Pi have the form D(Pi) x Ji(j) in Pi where Pj and Pt- are a 2-handle and a 1-handle respectively with j > i in a normal handle procedure P. For each 1-handle P,- in a normal handle procedure P , let Jt- be the union of the intervals J t (j) as in the previous sentence as j ranges over those integers from (z + 1 ) through the length of P for which Pj is a 2-handle. Let A{ be D(Pi) x (I - J,-) in D(Pi) x J = Pt-. Another view of Ai is that it is the closure of P,- minus the union of all 2-handles Pj in P for which j > i.