Universal abelian cover
= Universal abelian cover
{title2=$\widetilde Y_{\mathrm{ab}}$}
For a connected space, this is the connected covering corresponding to the kernel of the homomorphism from its <fundamental group> to its <abelianization>. Its <deck transformation group> is $H_1(Y;\mathbb Z)$. Lifted cells make its <cellular chain complex> a complex over the <group ring> $\mathbb Z[H_1(Y)]$.