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Transactions of the American Mathematical Society

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An algebraic determination of closed orientable $ 3$-manifolds


Authors: William Jaco and Robert Myers
Journal: Trans. Amer. Math. Soc. 253 (1979), 149-170
MSC: Primary 57N10
DOI: https://doi.org/10.1090/S0002-9947-1979-0536940-6
MathSciNet review: 536940
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Abstract: Associated with each polyhedral simple closed curve j in a closed, orientable 3-manifold M is the fundamental group of the complement of j in M, $ {\pi _1}(M - j)$. The set, $ \mathcal{K}(M)$, of knot groups of M is the set of groups $ {\pi _1}(M - j)$ as j ranges over all polyhedral simple closed curves in M. We prove that two closed, orientable 3-manifolds M and N are homeomorphic if and only if $ \mathcal{K}(M) = \mathcal{K}(N)$. We refine the set of knot groups to a subset $ \mathcal{F}(M)$ of fibered knot groups of M and modify the above proof to show that two closed, orientable 3-manifolds M and N are homeomorphic if and only if $ \mathcal{F}(M) = \mathcal{F}(N)$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0536940-6
Keywords: 3-manifold, knot-manifold, knot-group, fibered-knot, Seifert fibered manifold, characteristic Seifert manifold, characterization
Article copyright: © Copyright 1979 American Mathematical Society

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