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:
“We often overestimate the influence of a peer group on our teenager. While the peer group is most influential in matters of taste and preference, we parents are most influential in more abiding matters of standards, beliefs, and values.”
—David Elkind (20th century)
“One theme links together these new proposals for family policy—the idea that the family is exceedingly durable. Changes in structure and function and individual roles are not to be confused with the collapse of the family. Families remain more important in the lives of children than other institutions. Family ties are stronger and more vital than many of us imagine in the perennial atmosphere of crisis surrounding the subject.”
—Joseph Featherstone (20th century)
“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)
“Finally she grew quiet, and after that, coherent thought. With this, stalked through her a cold, bloody rage. Hours of this, a period of introspection, a space of retrospection, then a mixture of both. Out of this an awful calm.”
—Zora Neale Hurston (1891–1960)