The homology of cyclic and irregular dihedral coverings branched over homology spheres

Abstract

H. M. Hilden [Bull. Amer. Math. Soc. 80 (1974), 1243–1244; MR0350719 (50 #3211)], U. Hirsch [Math. Z. 140 (1974), 203–230] and Montesinos [Bull. Amer. Math. Soc. 80 (1974), 845–846] showed that every closed and orientable 3-manifold is a 3-fold dihedral covering space branched along a knot in S3. The purpose of the paper under review is to answer the question of whether this is true for irregular dihedral covering spaces branched over S3 with more than 3 sheets. The authors first show that for each odd prime p, the homology group Hi(M;Z) of every p-fold irregular dihedral covering space M over a homology n-sphere can be given the structure of a finitely generated module over the ring Z[ξ+ξ−1] of integers of the real cyclotomic field Q[ξ+ξ−1], where ξ=exp(2πi/p). For each odd prime p, using the fact that Z[ξ+ξ−1] is a Dedekind domain, they describe the class Dp of finitely generated abelian groups supporting the structure of a finitely generated module over Z[ξ+ξ−1], and prove that if M is a p-fold irregular dihedral covering space branched over a homology n-sphere, then Hi(M;Z)∈Dp, i≠0,n. This generalizes the results of Chumillas ["Study of dihedral coverings in S3 branched over knots'', Ph.D. Thesis, Madrid, 1984; per bibl.] and of A. Costa and J. M. Ruiz [Math. Ann. 275 (1986), no. 1, 163–168]. As a consequence of these results, they obtain 3-manifolds which are not p-fold irregular dihedral covering spaces branched over S3 for any prime p>3. The authors indicate that the method used in this paper is applicable to the case of cyclic covering spaces branched over a homology n-sphere. The realization problem (i.e., given a group G∈Dp, does there exist an irregular p-fold dihedral covering space p:M→S3 such that H1(M;Z) is isomorphic to G?) is also studied. Finally, the authors conclude by providing three tables which give the homology group of the p-fold irregular dihedral covering spaces of the knots of less than eleven crossings with more than 2 bridges for p=5,7,11.