Editing Covering group (section)

== Universal covering group ==
If ''H'' is a path-connected, locally path-connected, and [[semilocally simply connected]] group then it has a [[covering space#Universal covering|universal cover]]. By the previous construction the universal cover can be made into a topological group with the covering map a continuous homomorphism. This group is called the '''universal covering group''' of ''H''. There is also a more direct construction, which we give below.

Let ''PH'' be the [[path group]] of ''H''. That is, ''PH'' is the space of [[path (topology)|path]]s in ''H'' based at the identity together with the [[compact-open topology]]. The product of paths is given by pointwise multiplication, i.e. (''fg''){{nowrap|1=(''t'') = ''f''(''t'')''g''(''t'')}}. This gives ''PH'' the structure of a topological group. There is a natural group homomorphism {{nowrap|''PH'' → ''H''}} that sends each path to its endpoint. The universal cover of ''H'' is given as the quotient of ''PH'' by the normal subgroup of [[null-homotopic]] [[loop (topology)|loop]]s. The projection {{nowrap|''PH'' → ''H''}} descends to the quotient giving the covering map. One can show that the universal cover is [[simply connected]] and the kernel is just the [[fundamental group]] of ''H''. That is, we have a [[short exact sequence]]
: <math>1\to \pi_1(H) \to \tilde H \to H \to 1</math>
where {{overset|lh=0.6em|~|''H''}} is the universal cover of ''H''. Concretely, the universal covering group of ''H'' is the space of homotopy classes of paths in ''H'' with pointwise multiplication of paths. The covering map sends each path class to its endpoint.