Abstract
A finitely generated Abelian group G is a direct double if there exists H such that G = H ⊕ H. Let X̄ n be the n-sheeted cyclic covering of S 3 (or an integral homology sphere) branched over a knot. Plans' theorem states that H 1( X̄ n ; Z ) is a direct double if n is odd. This statement is wrong if the branching set is nonconnected. However: Theorem. Suppose that the branching set is a link. Then: 1. (1) If n is odd, the rank of H 1(X̄ n; Z ) is even and its p-torsion subgroup is a direct double if p n . 2. (2) If n = 2k, let Π : X̄ 2k → X̄ 2 be the obvious map. Then Ker Π ∗ has even rank and its p-torsion subgroup is a direct double if p k .
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