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:
“Jury—A group of twelve men who, having lied to the judge about their hearing, health, and business engagements, have failed to fool him.”
—H.L. (Henry Lewis)
“The structure was designed by an old sea captain who believed that the world would end in a flood. He built a home in the traditional shape of the Ark, inverted, with the roof forming the hull of the proposed vessel. The builder expected that the deluge would cause the house to topple and then reverse itself, floating away on its roof until it should land on some new Ararat.”
—For the State of New Jersey, U.S. public relief program (1935-1943)
“Three forms I see on stretchers lying, brought out there untended
lying,
Over each the blanket spread, ample brownish woolen blanket,
Gray and heavy blanket, folding, covering all.”
—Walt Whitman (1819–1892)
“But alas! I never could keep a promise. I do not blame myself for this weakness, because the fault must lie in my physical organization. It is likely that such a very liberal amount of space was given to the organ which enables me to make promises, that the organ which should enable me to keep them was crowded out. But I grieve not. I like no half-way things. I had rather have one faculty nobly developed than two faculties of mere ordinary capacity.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)