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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Inclusion maps of $ 3$-manifolds which induce monomorphisms of fundamental groups


Author: Jože Vrabec
Journal: Trans. Amer. Math. Soc. 214 (1975), 75-93
MSC: Primary 57A10
MathSciNet review: 0388398
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Abstract: The main result is the following ``duality'' theorem. Let M be a 3-manifold, P a compact and connected polyhedral 3-submanifold of $ \int M$, and X a compact and connected polyhedron in $ \int P$. If $ {\pi _1}(X) \to {\pi _1}(P)$ is onto, then $ {\pi _1}(M - P) \to {\pi _1}(M - X)$ is one-to-one. Some related results are proved, for instance: we can allow P to be noncompact if also X satisfies a certain noncompactness condition: if M lies in a 3-manifold W with $ {H_1}(W) = 0$, then the condition that $ {\pi _1}(X) \to {\pi _1}(P)$ is onto can be replaced by the weaker one that $ {H_1}(X) \to {H_1}(P)$ is onto.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0388398-4
PII: S 0002-9947(1975)0388398-4
Keywords: Inclusions of 3-manifolds, 1-acyclic 3-manifolds, fundamental groups of 3-manifolds
Article copyright: © Copyright 1975 American Mathematical Society