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

For a path-connected, locally path-connected, semilocally simply connected topological space $Y$, the universal abelian cover is the connected covering space associated to the commutator subgroup of $G={\pi }_1(Y,y_0)$. It is a regular covering with deck transformation groupEvery connected covering space of $Y$ with Abelian deck transformation group that is also a regular covering is a quotient of this cover: its corresponding normal subgroup contains $[G,G]$. This is universality among such abelian regular coverings, rather than a claim that the total space is simply connected.^[ref]

For a torus, $G\cong {\mathbb{Z}}^2$ is already Abelian, so this cover is the universal cover ${\mathbb{R}}^2$. For a wedge sum of two circles, $G$ is a free group on two generators and $[G,G]$ is nontrivial; the universal abelian cover is therefore not simply connected. If $Y$ is a CW complex, lifting its cells gives a cellular chain complex of free modules over the group ring $\mathbb{Z}[H_1(Y;\mathbb{Z})]$.