The homology of Abelian covers of knotted graphs
Let M̃ be a regular branched cover of a homology 3-sphere M with deck group G ≅ ℤd2 and branch set a trivalent graph Γ; such a cover is determined by a coloring of the edges of Γ with elements of G. For each index-2 subgroup H of G, MH = M̃/H is a double branched cover of M. Sakuma has proved that H1 (M̃) is isomorphic, modulo 2-torsion, to ⊕H H1(MH), and has shown that H1 (M̃) is determined up to isomorphism by ⊕H H1(MH) in certain cases; specifically, when d = 2 and the coloring is such that the branch set of each cover MH → M is connected, and when d = 3 and Γ is the complete graph K4. 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.
Publication Source (Journal or Book title)
Canadian Journal of Mathematics
Litherland, R. (1999). The homology of Abelian covers of knotted graphs. Canadian Journal of Mathematics, 51 (5), 1035-1072. https://doi.org/10.4153/CJM-1999-046-7