Total linking number modules

Abstract:

Given a codimension two link $L$ in a sphere ${S^k}$ with complement $X = {S^k} - L$, the total linking number covering of $L$ is the covering $\hat X \to X$ associated to the surjection ${\pi _1}(X) \to Z$ defined by sending the meridians to $1$. The homology ${H_{\ast }}(\hat X)$ define weaker invariants than the homology of the universal abelian covering of $L$. The groups ${H_i}(\hat X)$ are modules over $Z\left [ {t, {t^{ - 1}}} \right ]$ and this work gives an algebraic characterization of these modules for $k \geqslant 4$ except for the pseudo null part of ${H_1}(\hat X)$.