Page 159 - Curvature and Homology

4.5. CURVATURE AND BETTI NUMBERS 141

Now, by lemma 4.2.2, B(XP)wa = 0, a, = 1, --, n, that is the wa are
invariant. Hence, by the proof of prop. 4.4.3 they are harmonic with
respect to g. Since B(X), X E L is a derivation, B(X)& = 0 for any left
invariant p-form a. We conclude therefore that b,(G) = (i).
Theorem 4.4.2. A compact connected abelian Lie group G is a multi-torus.
To prove this we need only show that the vector fields X&(a = 1, -, n)
are parallel in the constructed metric. (This is left as an exercise for the
reader.) For, by applying the interchange formulae (1.7.19) to the
X,(u = 1, a*., n) and using the fact that the Xa are linearly independent
vector fields we conclude that G is locally flat. However, a compact
connected group which is locally isomorphic with En (as a topological
group) is isomorphic with the n-dimensional torus.
We have seen that the Euler characteristic of a torus vanishes. It is
now shown that for a compact connected semi-simple Lie group G,
x(G) = 0. Indeed, theaproof given is valid for any compact Lie group.
Let v, denote the number of linearly independent left invariant p-forms
no linear combination of which is closed; v,, is then the number
of linearly independent exact p-forms. Since the dimension of AP(T,*)
is (g) we have by the decomposition of a p-form

= (- lp+l v,, - Yo,
and so, since v, = v, = 0, x(G) = 0.

Theorem 4.4.3. The Euler characteristic of d compact connected Lie
group vanishes.

4.5. Curvature and betti numbers of a compact semi-simple Lie
group G

In this section we make use of the curvature properties of G in order
to prove theorem 4.4.1. We begin by forming the curvature tensor
defined by the connection (4.3.7). Denoting the components of this