Editing Covering group (section)

== Group structure on a covering space ==
Let ''H'' be a topological group and let ''G'' be a covering space of ''H''. If ''G'' and ''H'' are both [[path-connected]] and [[locally path-connected]], then for any choice of element ''e''* in the fiber over {{nowrap|''e'' ∈ ''H''}}, there exists a unique topological group structure on ''G'', with ''e''* as the identity, for which the covering map {{nowrap|''p'' : ''G'' → ''H''}} is a homomorphism.

The construction is as follows. Let ''a'' and ''b'' be elements of ''G'' and let ''f'' and ''g'' be [[path (topology)|path]]s in ''G'' starting at ''e''* and terminating at ''a'' and ''b'' respectively. Define a path {{nowrap|''h'' : ''I'' → ''H''}} by {{nowrap|1=''h''(''t'') = ''p''(''f''(''t''))''p''(''g''(''t''))}}. By the path-lifting property of covering spaces there is a unique lift of ''h'' to ''G'' with initial point ''e''*. The product ''ab'' is defined as the endpoint of this path. By construction we have {{nowrap|1=''p''(''ab'') = ''p''(''a'')''p''(''b'')}}. One must show that this definition is independent of the choice of paths ''f'' and ''g'', and also that the group operations are continuous.

Alternatively, the group law on ''G'' can be constructed by lifting the group law {{nowrap|''H'' × ''H'' → ''H''}} to ''G'', using the lifting property of the covering map {{nowrap|''G'' × ''G'' → ''H'' × ''H''}}.

The non-connected case is interesting and is studied in the papers by Taylor and by Brown-Mucuk cited below. Essentially there is an obstruction to the existence of a universal cover that is also a topological group such that the covering map is a morphism: this obstruction lies in the third cohomology group of the group of components of ''G'' with coefficients in the fundamental group of ''G'' at the identity.