Covering Group - Properties

Properties

Let G be a covering group of H. The kernel K of the covering homomorphism is just the fiber over the identity in H and is a discrete normal subgroup of G. The kernel K is closed in G if and only if G is Hausdorff (and if and only if H is Hausdorff). Going in the other direction, if G is any topological group and K is a discrete normal subgroup of G then the quotient map p : G → G/K is a covering homomorphism.

If G is connected then K, being a discrete normal subgroup, necessarily lies in the center of G and is therefore abelian. In this case, the center of H = G/K is given by

As with all covering spaces, the fundamental group of G injects into the fundamental group of H. If G is path-connected then the quotient group is isomorphic to K. Since the fundamental group of a topological group is always abelian, every covering group is a normal covering space. The group K acts simply transitively on the fibers (which are just left cosets) by right multiplication. The group G is then a principal K-bundle over H.

If G is a covering group of H then the groups G and H are locally isomorphic. Moreover, given any two connected locally isomorphic groups H₁ and H₂, there exists a topological group G with discrete normal subgroups K₁ and K₂ such that H₁ is isomorphic to G/K₁ and H₂ is isomorphic to G/K₂.