Page 158 - Curvature and Homology

140 IV. COMPACT LIE GROUPS

betti numbers of G has as a result been reduced to purely algebraic
considerations.
Remarks : In proving prop. 4.4.3 we obtained the formula

thereby showing that 6 is an anti-derivation in A(Tz). (The proposition
could have been obtained by an application of the Hodge-de Rham
decomposition of a form). It follows that the exterior product of harmonic
form on a compact semi-simple Lie group is ako harmonic.

Theorem 4.4.1. The Jirst and second betti numbers of a compact semi-
simple Lie group G vanish.
Let /? = Bawa be a harmonic 1-form. Then, from (4.4.2), B,, Gal$ = 0.
Multiplying these equations by CyMl = Cypal and contracting results
in By = 0, y = 1, .-., n.
If a = A4 wa A w@ is a harmonic 2-form, then by (4.4.2)

Permuting ar, /3 and y cyclically and adding the three equations obtained
gives
ApB Ca,P + ApO Cy$ + Ap y Cpap = 0,
and so from (4.4.3)

Multiplying these equations by Cd4 results in Ayd = 0 (y, S = 1, ..., n).
Suppose G is a compact but not necessarily semi-simple Lie group.
We have setn that the number of linearly independent left invariant
differential forms of degree p on G is (g). Tf we assume that bJG) = (g),
then the Euler characteristic x(G) of G is zero. For,

(This is not, however, a special implication of b,(G) = (g) (cf. theorem
4.4.3)).
A compact (connected) abelian Lie group G has these properties.
For, since G is abelian so is its Lie algebra L. Therefore, by (4.1.2)
its structure constants vanish. A metric g is defined on G as follows: