Cowsik and Swarup [CS] have shown that the homology groups of the infinite cyclic cover of a knot inject into the inverse limit of the homology groups of the branched cyclic covers. They also give conditions under which the injection is an isomorphism. We prove an analogous result for the fundamental group and generalize it to the case of links. The relationship between the homology of the infinite cyclic cover of a knot complement and that of the inverse limit of the branched cyclic covers has

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... been investigated in [CS, Dl, D2] . We would like to extend some of these results to homotopy. Let K be a knot in S3, A the infinite cyclic cover of A = S3 -K, and Xn the n-fold cyclic cover of A. We write £" for the branched n-fold cover of S3 branched over K. This gives the following commutative diagram: X -» Xk <-► Efc \ T T