Lorentz Group - Topology

Topology

The left and right groups in the double covering

SU(2) SO(3)

are deformation retracts of the left and right groups, respectively, in the double covering

SL(2,C) SO+(1,3).

But the homogeneous space SO+(1,3)/SO(3) is homeomorphic to hyperbolic 3-space H3, so we have exhibited the restricted Lorentz group as a principal fiber bundle with fibers SO(3) and base H3. Since the latter is homeomorphic to R3, while SO(3) is homeomorphic to three-dimensional real projective space RP3, we see that the restricted Lorentz group is locally homeomorphic to the product of RP3 with R3. Since the base space is contractible, this can be extended to a global homeomorphism.