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Transactions of the American Mathematical Society

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



Milnor’s invariants and the completions of link modules

Author: Lorenzo Traldi
Journal: Trans. Amer. Math. Soc. 284 (1984), 401-424
MSC: Primary 57M05; Secondary 57M25
MathSciNet review: 742432
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Abstract: Let $L$ be a tame link of $\mu \geqslant 2$ components in ${S^3}$, $H$ the abelianization of its group ${\pi _1}({S^3} - L)$, and $IH$ the augmentation ideal of the integral group ring ${\mathbf {Z}}H$. The $IH$-adic completions of the Alexander module and Alexander invariant of $L$ are shown to possess presentation matrices whose entries are given in terms of certain integers $\mu ({i_1}, \ldots ,{i_q})$ introduced by J. Milnor. Various applications to the theory of the elementary ideals of these modules are given, including a condition on the Alexander polynomial necessary for the linking numbers of the components of $L$ with each other to all be zero. In the special case $\mu = 2$, it is shown that the various Milnor invariants $\bar \mu ([r + 1,s + 1])$ are determined (up to sign) by the Alexander polynomial of $L$, and that this Alexander polynomial is $0$ iff $\bar \mu ([r + 1,s + 1]) = 0$ for all $r,s \geqslant 0$ with $r + s$ even; also, the Chen groups of $L$ are determined (up to isomorphism) by those nonzero $\bar \mu ([r + 1,s + 1])$ with $r + s$ minimal. In contrast, it is shown by example that for $\mu \geqslant 3$ the Alexander polynomials of a link and its sublinks do not determine its Chen groups.

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Keywords: Tame links, Milnor’s invariants, <IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$I$">-adic completions, Chen groups, elementary ideals, Alexander polynomials
Article copyright: © Copyright 1984 American Mathematical Society