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

 

Free coverings and modules of boundary links


Author: Nobuyuki Sato
Journal: Trans. Amer. Math. Soc. 264 (1981), 499-505
MSC: Primary 57Q45
MathSciNet review: 603777
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Abstract: Let $ L = \{ {K_1}, \ldots ,{K_m}\} $ be a boundary link of $ n$-spheres in $ {S^{n + 2}}$, where $ n \geqslant 3$, and let $ X$ be the complement of $ L$. Although most of the classical link invariants come from the homology of the universal abelian cover $ \tilde X$ of $ X$, with increasing $ m$ these groups become difficult to manage. For boundary links, there is a canonical free covering $ {X_\omega }$, which is simultaneously a cover of $ \tilde X$. Thus, knowledge of $ {H_ \ast }{X_\omega }$ yields knowledge of $ {H_ \ast }\tilde X$. We study general properties of such covers and obtain, for $ 1 < q < n/2$, a characterization of the groups $ {H_q}{X_\omega }$ as modules over the group of covering transformations. Some applications follow.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0603777-8
PII: S 0002-9947(1981)0603777-8
Article copyright: © Copyright 1981 American Mathematical Society