three strands. Lie groups The above definitions and constructions all apply to the special case of Lie groups. In particular, every covering of a manifold is a manifold, and the covering homomorphism becomes a smooth map. Likewise, given any discrete normal subgroup of a Lie group the quotient group is a Lie group and the quotient map is a covering homomorphism. Two Lie groups are locally isomorphic if and only if their Lie algebras are isomorphic. This implies that a homomorphism φ : G → H of Lie groups is a covering homomorphism if and only if the induced map on Lie algebras is an isomorphism. Since for every Lie algebra there is a unique simply connected Lie group G with Lie algebra , from this follows that the universal covering group of a connected Lie group H is the (unique) simply connected Lie group G having the same Lie algebra as H. Examples The universal covering group of the circle group T is the additive group of real numbers R with the covering homomorphism given by the exponential function exp: R → T. The kernel of the exponential map is isomorphic to Z. For any integer n we have a covering group of the circle by itself T → T which sends z to zn. The kernel of this homomorphism is the cyclic group consisting of the nth roots of unity. The rotation group SO(3) has as a universal cover the group SU(2) which is isomorphic to the group of versors in the quaternions. This is a double cover since the kernel has order 2. (cf the tangloids.) The unitary group U(n) is covered by the compact group T × SU(n) with the covering homomorphism given by p(z, A) = zA. The universal cover is R × SU(n). The special orthogonal group SO(n) has a double cover called the spin group Spin(n). For n ≥ 3, the spin group is the universal cover of SO(n). For n ≥ 2, the universal cover of the special linear group SL(n, R) is not a matrix group (i.e. it has no faithful finite-dimensional representations). References Topological groups Lie groups