Universal Covering Groups and Fundamental Groups Some Basic

Universal covering groups and fundamental groups

Some basic results related to universal covering groups of Lie groups will be summa- rized in class (without proofs, except possibly for item 2 below). Let G be a connected Lie group. Then, as a topological space, G has a universal covering space Ge. There are several results that could be studied for this topic. The first three items are proved in Warner’s book. Items 4 through 6 are related to material from Bump’s book, in Chapters 13 and 14. The approach there is to first show that local homomorphisms of topological groups can be extended when the domain is a simply connected topological group, and then to show how to lift a Lie algebra homomorphism to a local homomorphism of Lie groups. This topic might involve learning proofs of items 1-3 or else 4-6, and then adding some material from items 7 and 8. There may be other references that might be useful for this topic. 1. The space Ge has a unique differentiable structure for which the covering map π : Ge → G is smooth non-singular map. There is a group structure on Ge making Ge into a Lie group and making π into a Lie group homomorphism. 2. Let G and H be connected Lie groups, and let ϕ : G → H be a homomorphism. Then

ϕ is a covering map if and only if dϕ : T1(G) → T1(H) is an isomorphism. 3. If G and H are Lie groups, G is simply connected, and ψ : g → h is a Lie algebra homomorphism, then there exists a (unique) homomorphism ϕ : G → H such that dϕ = ψ. (Here, g and h are the Lie algebras of G and H, respectively.) 4. If G and H are topological groups and G is simply connected, then any local homomorphism from a neighbourhood of 1 in G to H can be extended to a homomorphism from G to H. 5. If G and H are Lie groups and ψ : g → h is a Lie algebra homomorphism, then there exists a local homomorphism from an open neighbourhood of 1 in G to H. 6. Combining 4 and 5 to prove the result of 3. 7. Material from Chapter 13 of Bump. Computing fundamental groups of some noncom- pact Lie groups (using the Cartan decomposition). (Note: This would not involve proving the Cartan decomposition - that will be covered in class.) 8. Material from Chapter 22 of Bump, re the fundamental group of a compact Lie group.