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Compact submanifolds of 3-manifolds with boundary. (English) Zbl 0628.57008
A basic tool in 3-manifold theory is the result of Peter Scott that a 3- manifold, Y, with \(\pi_ 1(Y)\) finitely generated has a core - a compact submanifold, X, of Y such that \(i_*: \pi_ 1(X)\to \pi_ 1(Y)\) is an isomorphism. This paper extends this result to show that if C is a compact submanifold of \(\partial Y\) then X may be chosen so that \(X\cap \partial Y=C\). Some corollaries which follow: (1) Any compact, incompressible surface in Y is contained in some core of Y (an example shows that this is false for compressible surfaces), (2) there is a core, X, for Y such that \(\partial Y-X\) is planar, (3) there is a core, X, of Y such that for each component, F, of \(\partial Y\) with \(F\neq {\mathbb{R}}^ 2,S^ 2\), \(S^ 1\times {\mathbb{R}}\), \(X\cap F\) is a core of F.
Reviewer: J.Hempel

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
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