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:
“Stripped of ethical rationalizations and philosophical pretensions, a crime is anything that a group in power chooses to prohibit.”
—Freda Adler (b. 1934)
“Who says that fictions only and false hair
Become a verse? Is there in truth no beauty?
Is all good structure in a winding stair?
May no lines pass, except they do their duty
Not to a true, but painted chair?”
—George Herbert (15931633)
“You had to have seen the corpses lying there in front of the schoolthe men with their caps covering their facesto know the meaning of class hatred and the spirit of revenge.”
—Alfred Döblin (18781957)
“Sir Walter Raleigh might well be studied, if only for the excellence of his style, for he is remarkable in the midst of so many masters. There is a natural emphasis in his style, like a mans tread, and a breathing space between the sentences, which the best of modern writing does not furnish. His chapters are like English parks, or say rather like a Western forest, where the larger growth keeps down the underwood, and one may ride on horseback through the openings.”
—Henry David Thoreau (18171862)