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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Torsion-groups of abelian coverings of links


Authors: John P. Mayberry and Kunio Murasugi
Journal: Trans. Amer. Math. Soc. 271 (1982), 143-173
MSC: Primary 57M25; Secondary 57M12
MathSciNet review: 648083
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Abstract: If $ M$ is an abelian branched covering of $ {S^3}$ along a link $ L$, the order of $ {H_1}(M)$ can be expressed in terms of (i) the Alexander polynomials of $ L$ and of its sublinks, and (ii) a "redundancy" function characteristic of the monodromy-group. In 1954, the first author thus generalized a result of Fox (for $ L$ a knot, in which case the monodromy-group is cyclic and the redundancy trivial); we now prove earlier conjectures and give a simple interpretation of the redundancy. Cyclic coverings of links are discussed as simple special cases.

We also prove that the Poincaré conjecture is valid for the above-specified family of $ 3$-manifolds $ M$.

We state related results for unbranched coverings.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0648083-1
Keywords: Knots, links, unbranched coverings, branched coverings, abelian coverings, monodromy-groups, Seifert matrices, elementary divisors, Alexander matrix, Alexander polynomial, Poincaré conjecture, homology spheres, redundancy of a covering, $ 3$-manifolds, cyclic covering
Article copyright: © Copyright 1982 American Mathematical Society