Universal Covering Group
If H is a path-connected, locally path-connected, and semilocally simply connected group then it has a 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 paths in H based at the identity together with the compact-open topology. The product of paths is given by pointwise multiplication, i.e. (fg)(t) = f(t)g(t). This gives PH the structure of a topological group. There is a natural group homomorphism PH → H which 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 loops. The projection 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
where 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.
Read more about this topic: Covering Group
Famous quotes containing the words universal, covering and/or group:
“The Universal Soul, as it is called, has an interest in the stacking of hay, the foddering of cattle, and the draining of peat-meadows.”
—Henry David Thoreau (1817–1862)
“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)
“Just as a person who is always asserting that he is too good-natured is the very one from whom to expect, on some occasion, the coldest and most unconcerned cruelty, so when any group sees itself as the bearer of civilization this very belief will betray it into behaving barbarously at the first opportunity.”
—Simone Weil (1910–1943)