The Homology of Abelian Covers of Knotted Graphs

Litherland, R. A.

Canadian journal of mathematics, Tome 51 (1999) no. 5, pp. 1035-1072

Résumé

Let $\tilde{M}$ be a regular branched cover of a homology 3-sphere $M$ with deck group $G\cong \mathbb{Z}_{2}^{d}$ 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 $G,\,{{M}_{H}}=\tilde{M}/H$ is a double branched cover of $M$ . Sakuma has proved that ${{H}_{1}}\left( {\tilde{M}} \right)$ is isomorphic, modulo 2-torsion, to ${{\oplus }_{H}}{{H}_{1}}\left( {{M}_{H}} \right)$ , and has shown that ${{H}_{1}}\left( {\tilde{M}} \right)$ is determined up to isomorphism by ${{\oplus }_{H}}{{H}_{1}}\left( {{M}_{H}} \right)$ in certain cases; specifically, when $d=2$ and the coloring is such that the branch set of each cover ${{M}_{H}}\to M$ 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 $d=3\,\text{or}\,\text{4}$ , for an infinite class of colored graphs; and when $d=5$ , for a single coloring of the Petersen graph.

Détail

DOI : 10.4153/CJM-1999-046-7

Mots-clés : 57M12, 57M25, 57M15

Bibliographie

[1] [1] Flapan, Erica, Symmetries of Möbius ladders. Math. Ann. 283(1989), 271–283. Google Scholar

[2] [2] Kinoshita, Shin’ichi, On the three-fold irregular branched coverings of spatial four-valent graphs and its applications. J. Math. Chem. 14(1993), 47–55. Google Scholar

[3] [3] Lee, Ronnie and Weintraub, Steven H., On the homology of double branched covers. Proc. Amer. Math. Soc. 123(1995), 1263–1266. Google Scholar

[4] [4] Massey, W. S., Completion of link modules. Duke Math. J. 47(1980), 399–420. Google Scholar

[5] [5] Sakuma, Makoto, Homology of abelian coverings of links and spatial graphs. Canad. J. Math. (1) 47(1995), 201–224. Google Scholar

[6] [6] Simon, Jonathan, A topological approach to the stereochemistry of nonrigid molecules. Graph theory and topology in chemistry (Athens, Ga. 1987), Elsevier, Amsterdam-New York, 1987, 43–75. Google Scholar

[7] [7] Watkins, Mark E., A theorem on Tait colorings with an application to the generalized Petersen graphs. J. Combin. Theory 6(1969), 152–164. Google Scholar

[8] [8] Zariski, Oscar and Samuel, Pierre, Commutative Algebra, Vol. II. Springer-Verlag, New York, 1975; originally published by Van Nostrand, Princeton, NJ, 1960. Google Scholar