Free coverings and modules of boundary links

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.