Page 150 - Curvature and Homology

CHAPTER IV

COMPACT LIE GROUPS

The results of the previous chapter are now applied to the problem
of determining the betti numbers of a compact semi-simple Lie group G.
On the one hand, we employ the facts on curvature and betti numbers
already established, and on the other hand, the theory of invariant
differential forms. It turns out that the harmonic forms on G are precisely
those differential forms invariant under both the left and right trans-
lations of G. The conditions of invariance when expressed analytically
reduce the problem of the determination of betti numbers to a purely
algebraic one. No effort is made to compute the betti numbers of the
four main classes of simple Lie groups since this discussion is beyond
the scope of this book. However, for the sake of completeness, we give
the PoincarC polynomials in these cases omitting those for the five
exceptional simple Lie groups.
Locally, G has the structure of an Einstein space of positive curvature
and this fact is used to prove that the first and second betti numbers
vanish. These results are also obtained from the theory of invariant
differential forms. The existence of a harmonic 3-form is established
from differential geometric considerations and this fact allows us to
conclude that the third betti number is greater than or equal to one.
It is also shown that the Euler-PoincarC characteristic is zero.

4.1. The Grassman algebra of a Lie group

Consider a compact (connected) Lie group G. Its Lie algebra L
has as underlying vector space the tangent space Te at the identity e E G.
We have seen (§ 3.6) that an element A E Te determines a unique left
invariant infinitesimal transformation which takes the value A at e;
moreover, these infinitesimal transformations are the elements of L.
Let Xa(a = 1, ..a, n) be a base of the Lie algebra L and oa(a = 1, ..., n)
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