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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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Keywords: Inclusions of 3-manifolds, 1-acyclic 3-manifolds, fundamental groups of 3-manifolds
Article copyright: © Copyright 1975 American Mathematical Society