CHAPTER 21

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 α ^{p} is a bi-invariant p-form, then α is closed*,

**PROOF:** Let *σ*^{1},…, *σ ^{n}* and

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