Page 163 - Curvature and Homology

EXERCISES

EXERCISES

A. The second betti number of a compact semi-simple Lie group
1. Prove that b,(G) = 0 by showing that if a is an harmonic 2-form, then i(X)a
vanishes for any X EL. Make use of the fact that b,(G) = 0.

B. The third betti number of a compact simple Lie group [48]
1. Let Q(L) denote the vector space of invariant bilinear symmetric forms on L,
that is, the space of those forms q such that

for any X,Y,Z E L. To each q E Q(L) we associate a 3-form a(q) by the condition

Evidently, the map
4 -+ 44)
is linear.
2. For each q E Q(L) show that a(q) is invariant, and hence harmonic.
3. Since the derived algebra [L,L] = {[x,Y] I X,Y EL) coincides with L,
the map q -t a(q) of
QtL) + A: (T*)

is an isomorphism into. Show that it is an isomorphism onto. Hence, b,(G) =
dim Q(L).
Hint: For any element a E A~(T*) and X E L, i(X)a is closed. Since b,(G) = 0,
there is a 1-form /3 = Px such that i(X)a = dPs Now, show that

that is

Finally, show that the bilinear function
qtX,Y) = - (X, BY)
is invariant.
4. Prove that if G is ;r simple Lie group, then b,(G) = 1.