Universal abelian cover (${\tilde{Y}}_{\mathrm{ab}}$)

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 $\mathbb{Z}[H_1(Y)]$.