Finished!
On a compact, connected Lie group, every closed differential form is cohomologous to a left-invariant form, and you can compute the cohomology groups using only the left-invariant subcomplex, which is precisely the exterior algebra of the (dual of the) Lie algebra, and thus amenable to completely algebraic treatment.
Topology=Algebra.
And I did it in a cute little way, too! With respect to the unique bi-invariant Riemannian metric on the group, the harmonic forms all turn out to be left-invariant. QE freaking D, baby.
...Unless I'm wrong, of course.
Oddly enough, I got a lot of my inspiration from Samuel Goldberg's Curvature and Homology, despite the fact that I couldn't even read most of his coordinate-heavy proofs. (Though perhaps 'couldn't be bothered to try and read' would be more accurate. Coordinates are the Devil's assgoblins.)
Posted by aloysius at April 18, 2004 07:17 PM | TrackBack |