Betti Numbers and Covering Spaces
21.1. Bi-invariant Forms on Compact Groups
Why is it that the 1-parameter subgroups of a compact Lie group are geodesics?
Samelson’s article [Sam] is a beautiful exposition on the topology of Lie groups as it was known up to 1951.
21.1a. Bi-invariant p-Forms
Recall that a form or vector field on G is said to be bi-invariant if it is both left and right invariant. For example, on the affine group G = A(1) of the line, dx/x is bi-invariant.
Theorem (21.1): If αp is a bi-invariant p-form, then α is closed,
PROOF: Let σ1,…, σn and τ1,…, τn be bases of the left and the right invariant 1-forms, respectively, ...