Projective Linear Group - Topology

Topology

Over the real and complex numbers, the topology of PGL and PSL can be determined from the fiber bundles that define them:

via the long exact sequence of a fibration.

For both the reals and complexes, SL is a covering space of PSL, with number of sheets equal to the number of nth roots in K; thus in particular all their higher homotopy groups agree. For the reals, SL is a 2-fold cover of PSL for n even, and is a 1-fold cover for n odd, i.e., an isomorphism:

For the complexes, SL is an n-fold cover of PSL.

For PGL, for the reals, the fiber is so up to homotopy, is a 2-fold covering space, and all higher homotopy groups agree.

For PGL over the complexes, the fiber is so up to homotopy, is a circle bundle. The higher homotopy groups of the circle vanish, so the homotopy groups of and agree for In fact, always vanishes for Lie groups, so the homotopy groups agree for