The homology of Abelian covers of knotted graphs

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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.

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Canadian Journal of Mathematics

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