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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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