Let $ ilde M$ be a regular branched cover of a homology 3-sphere$M$ with deck group $Gcong zt^d$ 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$G$, $M_H = ilde M/H$ is a double branched cover of $M$. Sakumahas proved that $H_1( ilde M)$ is isomorphic, modulo 2-torsion, to$igoplus_H H_1(M_H)$, and has shown that $H_1( ilde M)$ isdetermined up to isomorphism by $igoplus_H H_1(M_H)$ in certaincases; specifically, when $d=2$ and the coloring is such that thebranch set of each cover $M_H o M$ is connected, and when $d=3$and $Gamma$ is the complete graph $K_4$. We prove this for alarger class of coverings: when $d=2$, for any coloring of aconnected graph; when $d=3$ or $4$, for an infinite class ofcolored graphs; and when $d=5$, for a single coloring of thePetersen graph.
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机译：假设$ ilde M $是同构3球体$ M $的正规分支覆盖，其中甲板组$ Gcong zt ^ d $，分支设置了三价图$ Gamma $;这样的覆盖由$ Gamma $的边缘用$ G $元素着色决定。对于$ G $的每个索引2子组$ H $，$ M_H = ilde M / H $是$ M $的双分支覆盖。 Sakumahas证明$ H_1（ilde M）$是同构的，模2扭转的，等于$ igoplus_H H_1（M_H）$，并且证明$ H_1（ilde M）$由$ igoplus_H H_1（M_H）$决定为同构。在某些情况下具体来说，当$ d = 2 $并且着色使得每个封面$ M_H o M $的分支集相连时，并且当$ d = 3 $和$ Gamma $是完整图形$ K_4 $时。对于更大范围的覆盖，我们证明了这一点：当$ d = 2 $时，对于连通图的任何着色;当$ d = 3 $或$ 4 $时，表示无限类的彩色图形;当$ d = 5 $时，为Petersen图的单一着色。
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