If $M$ is an abelian branched covering of $S^3$ along a link $L$, the order of $H_1(M)$ can be expressed in terms of (i) the Alexander polynomials of $L$ and of its sublinks, and (ii) a "redundancy" function characteristic of the monodromy-group. In 1954, the first author thus generalized a result of Fox (for $L$ a knot, in which case the monodromy -group is cyclic and the redundancy trivial); we now prove earlier conjectures and give a simple interpretation of the redundancy. Cyclic coverings of links are discussed as simple special cases. We also prove that the Poincaré conjecture is valid for the above-specified family of 3-manifolds $M$. We state related results for unbranched coverings.