Every lie group has a maximal covering group. A covering group is a group with the same lie algebra that contains all other lie groups with the same lie algebra.

Let G1,..,Gn with the same lie algebra. There exists G, a universal covering group with G->Gi of finite order.

G is the only simply connected group.

Example:
SU(2) covers SO(3)

Universal covering group of SO(m)= Spin(m) Spin(m) has spinor reps not reps of SO(m) Spin(4)~=SU(2)xSU(2) with (j1,j2) = eigenvalues. SO(4) satisfy j1+j2= integer.

=> Spin(4) -> SO(4)-> SO(3)xSO(3) with each mapping 2->1.