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 e ∈ H, there exists a unique topological group structure on G, with e* as the identity, for which the covering map 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 paths in G starting at e* and terminating at a and b respectively. Define a path h : I → H by 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 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.
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 which 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.
Read more about this topic: Covering Group
Famous quotes containing the words group, structure, covering and/or space:
“The virtue of dress rehearsals is that they are a free show for a select group of artists and friends of the author, and where for one unique evening the audience is almost expurgated of idiots.”
—Alfred Jarry (1873–1907)
“The verbal poetical texture of Shakespeare is the greatest the world has known, and is immensely superior to the structure of his plays as plays. With Shakespeare it is the metaphor that is the thing, not the play.”
—Vladimir Nabokov (1899–1977)
“You had to have seen the corpses lying there in front of the school—the men with their caps covering their faces—to know the meaning of class hatred and the spirit of revenge.”
—Alfred Döblin (1878–1957)
“Not so many years ago there there was no simpler or more intelligible notion than that of going on a journey. Travel—movement through space—provided the universal metaphor for change.... One of the subtle confusions—perhaps one of the secret terrors—of modern life is that we have lost this refuge. No longer do we move through space as we once did.”
—Daniel J. Boorstin (b. 1914)