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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)

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M05 Fundamental group, presentations, free differential calculus
##### Keywords:
cores of noncompact 3-manifolds; irreducible core
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