The homology of abelian coverings of knotted graphs

Abstract

Let N be a regular branched cover of a homology 3-sphere M with deck group G isomorphic to Z_2^d and branch set a trivalent graph Gamma; such a cover is determined by a coloring of the edges of Gamma with elements of G. For each index-2 subgroup H of G, M_H = N/H is a double branched cover of M. Sakuma has proved that the first homology of N is isomorphic, modulo 2-torsion, to the direct sum of the first homology groups of the M_H, and has shown that H_1(N) is determined up to isomorphism by the direct sum of the H_1(M_H) in certain cases; specifically, when d=2 and the coloring is such that the branch set of each cover M_H -> M is connected, and when d=3 and Gamma is the complete graph K_4. We prove this for a larger class of coverings: when d=2, for any coloring of a connected graph; when d=3 or 4, for an infinite class of colored graphs; and when d=5, for a single coloring of the Petersen graph.