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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A note on the construction of simply-connected $ 3$-manifolds as branched covering spaces of $ S\sp{3}$

Author: Joan S. Birman
Journal: Proc. Amer. Math. Soc. 55 (1976), 440-442
MathSciNet review: 0394629
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Abstract: Let $ K$ be a knot in $ {S^3}$, and let $ \omega :{\pi _1}({S^3} - K) \to {\Sigma _n}$ be a transitive representation into the symmetric group $ {\Sigma _n}$ on $ n$ letters. The pair $ (K,\omega )$ defines a unique closed, connected orientable $ 3$-manifold $ M(K,\omega )$, which is represented as an $ n$-sheeted covering space of $ {S^3}$, branched over $ K$. A procedure is given for representing $ M(K,\omega )$ by a Heegard splitting, and a formula is given for computing the genus of that Heegard splitting of $ M(K,\omega )$. This formula is then applied to the $ 3$-sheeted irregular covering spaces studied by Hilden (Bull. Amer. Math. Soc. 80 (1974), 1243-1244) and Montesinos (Bull. Amer. Math. Soc. 80 (1974), 845-846), and, also, Tesis (Univ. de Madrid, 1971) to show that these particular covering spaces cannot yield counterexamples to the Poincaré Conjecture if the branch set has bridge number $ < 4$.

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Keywords: $ 3$-manifold, homotopy sphere, branched covering space, knot, Heegard splitting
Article copyright: © Copyright 1976 American Mathematical Society