Simply connected Lie group

A Lie group $G$ is simply connected if, as a topological space (equivalently a manifold), it is connected and has trivial fundamental group:

In particular, $G$ is a connected Lie group .

Simply connected Lie groups play a special role because they are the “global objects” most faithfully represented by Lie algebra data. Concretely:

• Every connected Lie group has a universal covering group $\widetilde G\to G$ whose total space $\widetilde G$ is a simply connected Lie group (see existence of universal covering groups and covering Lie group ).
• The Lie algebra does not see discrete topology: $G$ and $\widetilde G$ have the same Lie algebra (see Lie algebra of a Lie group ), but different global topology.

A key payoff is the uniqueness principle simply connected determined by Lie algebra : connected simply connected Lie groups are determined up to isomorphism by their Lie algebras, and Lie algebra homomorphisms integrate uniquely to group homomorphisms in the simply connected setting.