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Uniqueness of cores of noncompact 3-manifolds. (English) Zbl 0556.57009
Peter Scott proved that if M is a 3-manifold with finitely generated fundamental group, then there is a compact connected 3-manifold N in M- \(\partial M\) such that the inclusion \(i: N\to M\) induces an isomorphism \(i_{\#}: \pi_ 1(N)\to \pi_ 1(M)\). Such an N is called a core of M, and if N is irreducible, an irreducible core. We prove the following uniqueness theorem: Let M be a \(P^ 2\)-irreducible noncompact 3-manifold with finitely generated fundamental group. Suppose \(N_ 1\) and \(N_ 2\) are irreducible cores of M. Then there is a homeomorphism from \(N_ 1\) to \(N_ 2\). This homeomorphism cannot always be chosen to extend to M, but it can be made canonical in the following algebraic sense: Theorem: Let \(i_ 1: (N_ 1,x_ 1)\to (M,x)\) and \(i_ 2: (N_ 2,x_ 2)\to (M,x)\) be inclusions of two irreducible cores into the \(P^ 2\)- irreducible 3-manifold M. Then there is a homeomorphism h: (N\({}_ 1,x_ 1)\to (N_ 2,x_ 2)\) such that \(h_{\#}=(i_ 2)_{\#}^{- 1}(i_ 1)_{\#}: \pi_ 1(N_ 1,x_ 1)\to \pi_ 1(N_ 2,x_ 2)\).
Reviewer: Reviewer (Berlin)

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
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