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

By Matthew G. Brin

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Example text

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. The derived l-handles A(P) of P are the components of 22 MATTHEW G. BRIN AND T. L. THICKSTUN all the A{ for those i where P,- is a 1-handle of P. Note that if P,- and Pj are intersecting 1-handles in P with j > i, then there must be a set of 2-handles in P with indices between i and j whose union contains Pj n Pj. Thus A{ C\Aj = 0 whenever i ^ j and both A,- and Aj are defined.

4. END 1-MOVABILITY OF INTERIORS The analysis of end 1-movable 3-manifolds in the case of compact boundary is easier than the analysis in the case considered by this paper. See [BT1]. It is also easier in the case where the boundary is of finite type — where the boundary is the interior of a compact 2-manifold. One can obtain a manifold whose boundary has finite type from an arbitrary non-compact 3-manifold by removing some of the boundary. 1) that if a non-compact 3-manifold is end 1-movable, then it remains end 1-movable if a portion of the boundary is removed so that the remaining boundary has finite type.

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. Then there is a compact, connected submanifold M of V, with K C M, so that Fr M is incompressible in V — K, and so that each complementary domain W of M in V is unbounded, has connected frontier and 38 MATTHEW G. BRIN AND T. L. THICKSTUN satisfies one of the following: I. The homomorphism 7TIFTW —> niW is an isomorphism. II. The homomorphism %IFTW —> TTIW is one to one but not onto, and for any connected exhaustion (Mi), i > 0, of V with Mo — M, with each component of V — Mi unbounded for all i, and with each FT Mi, i > 0, incompressible in V — M, there is, for each i > 0, a unique component Fi of FT Mi so that Fi separates Fi+i from FTW in W, and so that if Ai and Bi are the components of W — Fi with FTW in FrAi, then KiFi —+ 7Ti Ai is an isomorphism, and 7TiF,- —• wiBi is not onto.