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

By Matthew G. Brin

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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 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).

The above arguments also show that no other component of V — M,- 40 MATTHEW G. BRIN AND T. L. THICKSTUN in W has this property. If we now redefine the symbols Fi so that, for each i, Fi denotes Fr W[' where W[' is the unique component of V — Mi in W with 71*1 FrWi —• K\Wi not onto, then the above arguments show that the (F{) also satisfy item II of the conclusion. 2 and are left to the reader. 4: The proof, except for some worry about connected summands, is almost identical to the proofs of Lemmas 1 and 2 of [Tu], and we will appeal to [Tu] for some important details.

We know that a pushes to the ends of V in V — K. The domain of this homotopy is the half open annulus S1 x [0,oo). Since FrJVo is incompressible in V — K > we can assume that the preimage of FriVo contains only essential circles in S1 x [0, oo). Either the preimage of Fr N0 is empty, and a pushes to the ends of V in V — N0; or the preimage of FriVo has an essential circle in S1 x [0,oo) and a homotops into Fr iVo in V — No. In either case a can be homotoped off any compact set in V-N0. We will now choose iVo so that every component of V — iVo satisfies either item I or item II of the conclusion.