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A branched covering is a map that is a covering space away from a branch set and has a specified local covering model at that set. For a smooth two-fold covering branched over a codimension-two submanifold, the local model is $(z,u)\mapsto (z^2,u)$. A two-fold branched cover of a knot is the basic three-dimensional example.

A covering homomorphism is a surjective homomorphism of topological groups whose underlying map is a covering space. Its discrete kernel is the fiber over the identity.

The connected covering space corresponding to $\ker \alpha$ has deck transformation group $\mathbb{Z}$. Choosing a deck generator makes its integral homology a module over the Laurent polynomial ring $\mathbb{Z}[t,t^{-1}]$.

For $C_2\ast \mathbb{Z}\to S_3$ taking the two generators to a transposition and a $3$-cycle, the kernel's six-sheeted regular covering space consists of three spheres and two circles. Each sphere has two marked points, and each circle meets one point on each sphere. It retracts onto a rank-four graph by replacing each sphere by a meridian arc, but admits no deformation retraction onto any graph because its degree-two homology is ${\mathbb{Z}}^3$.