On the homology of metacyclic coverings

Abstract

In this paper the authors study irregular metacyclic branched covering spaces. These arise as follows: Suppose G is a Z/m extension of Z/n. Then G contains a cyclic subgroup of order m, Cm, which we suppose is not normal. Suppose G acts on a PL manifold X. Then there are maps X→X/Cm→X/G. The map M=X/Cm→X/G=S is an irregular metacyclic covering. (It is not induced by the action of a group on M because Cm is not normal.) In the most interesting case S is a simply connected manifold, often a sphere. Then M→S is a covering space in the usual sense away from a codimension 2 subset of S, called the branch set. Usually the branch set is taken to be a knot or link in S. Regular branched coverings have been widely studied in many contexts (knot theory, algebraic geometry, etc.), but irregular coverings have been much less studied although they are important also. For example, every closed oriented 3-manifold is a 3-fold irregular covering of S3 with branch set a knot. (This is the case G equals the dihedral group of order 6.). That this result does not generalize to dihedral groups of order 2p, p an odd prime, follows from Theorem 2 of this paper, which is too technical to state here. But specializing Theorem 2 to the context of rational homology and dihedral groups of order 2p, p an odd prime, the authors obtain the following theorem: dimH1(M;Q)≡0 modulo 1/2(p−1). The methods of proof in the paper are algebraic.