The homology of abelian coverings of knotted graphs

1998/05/12 by R. A. Litherland, Litherland, R. A.
Computer Science · Mathematics · #57M12 (Primary) 57M25 #57M15 (Secondary #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GT #msc:57M12 #msc:57M15 #msc:57M25

paper · pdf · doi:10.48550/arxiv.math/9805058

Plain TeX, 40 pages, 1 figure; corrected the proof of Theorem 8.8, and some typos

openalex publication_date 1998/05/12 · arxiv created 1998/07/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let N be a regular branched cover of a homology 3-sphere M with deck group G isomorphic to Z₂^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 H₁(N) is determined up to isomorphism by the direct sum of the H₁(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 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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