Page 162 - Curvature and Homology

144 IV. COMPACT LIE GROUPS

2) The group B,: this is the orthogonal group in (21 + 1)-space the
elements of which have determinant + 1 ;
3) The group C,: this is the symplectic group in 21-space, that is C,
is the group of unitary transformations leaving invariant the skew-
symmetric bilinear form Z&, auxtyi where the coefficients are given by
a,-,,, - = 1 with all other a,* = 0;
- - a ,,,-I
4) The group Dl of orthogonal transformations in 21-space (1 = 3,
4, *em), the elements of which have determinant + 1.

There are also five exceptional compact simple Lie groups whose
dimensions are 14, 52, 78, 133, and 248 commonly denoted by G,,
F,, E,, E,, and E,, respectively.
The polynomial pG(t) = b, + b,t + + b,tn where the b4 (i =
0, -a, n) are the betti numbers of G is known as the Poincard polynomial
of G. Let G = G, x x G, where the Gd (i = 1, ..., k) are simple.
Then, it can be shown that

where pGI(t) is the PoincarC polynomial of G,. Therefore, in order to
find the betti numbers of a compact semi-simple Lie group we first
express it as the direct product of simple Lie groups, and then compute
the PoincarC polynomials of these groups, after which we employ the
formula (4.6.1).
Regarding the topology of a compact simple Lie group we already
know that (a) it is orientable; (b) b, = b, = 0, b3 2 1 and, therefore,
since the star operator is an isomorphism (or, by PoincarC duality)
bn-, = bn-, = 0, b,-, 2 1 ; (c) the Euler characteristic vanishes.
We conclude this chapter by giving (without proof) the PoincarC
polynomials of the four main classes of simple Lie groups:

Remark : A, = B1 = C1, B, = C2 and A3 = Ds.