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The Geometry of Physics by Theodore Frankel

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CHAPTER 21

Betti Numbers and Covering Spaces

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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, ...

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