Milnor’s invariants and the completions of link modules

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.