Covering homomorphism

A covering homomorphism is a surjective group homomorphism between topological groups whose underlying map is a covering map. Its group kernel is the discrete fibre above the identity element. Group translations identify every other fibre with a coset of that group kernel.

For example, the group homomorphism $\mathbb{R}\to S^1$, $t\mapsto e^{2\pi it}$, is a covering map from the additive real numbers to the circle group, with group kernel $\mathbb{Z}$. Small open arcs lift to disjoint intervals, while adding an integer moves between the sheets. A surjective group homomorphism alone need not be a covering map; the local-sheet condition is additional topological information.