The homology of cyclic branched covers of S 3

Given a knot K in S 3 and a positive integer p, there is a unique p-fold cyclic connected cover X v --, S 3 K, and this can be completed to a branched cover M e --* S 3. When p is prime, the homology group H1 (M e) is torsion and was one of the earliest knot invariants (predating the Alexander polynomial). It was used by Alexander and Briggs [A-B] to distinguish knots up to 8 crossings and all but 3 pairs of 9 crossing knots, in the first systematic at tempt to distinguish knots. This invariant was discovered independently by Reidemeister [R]. The basic goal of this paper is to determine the range of this invariant. It is well known (see e.g. [G]) that HI(Mv;F t , ) ) = 0 for a prime p. For p odd, A. Plans [P] showed that HI(Mp) is a direct double, isomorphic as an abelian group to G @ G. Further restrictions are given by the observation of Fox I F ] that for a prime p, HI(Mp) is in a natural way a Z [(]-module, where ( = exp(27zi/p) is a primitive p-th root of 1. Indeed, the branched cover is specified by an action of the finite cyclic group Cp on a closed 3-manifold with quotient S 3, and a transfer argument shows that the norm element S = 1 + 9 + 9 9 9 + gP-~ annihilates HI(Me) , hence HI(Mp) is a Z [ C v ] / 2; = Z [(]-module. Algebraic number theory establishes which abelian groups can be given the structure of a Z [(]-module; this was worked out in [C-M], see also [D]. This paper determines what abelian groups can arise as H1 (Mp) by giving necessary and sufficient conditions for a f.g. (finitely generated) Z [ ( ] -module to be isomorphic to H I ( M ) for some p-fold cyclic cover M ~ S 3 branched over a knot. The new feature is the role of the ideal class group C(Z [ ( ] ) of the Dedekind domain Z [(]. For any f.g. Z [(]-module Q, there is an ideal class