Characteristic classes for $GO(2N, \mathbb{C})$
Yogish I. Holla, Nitin Nitsure · Asian Journal of Mathematics · 2001
The complex Lie group GO(n) is by definition the closed subgroup of GL(n) consisting of all matrices g such that t gg is a scalar matrix \I for some A G (E*. (We write simply GL(n), SO(n), O(n), etc. for the complex Lie groups GL{n,(U), SO{n,W), 0(n, 3, the group GO(2n -j-1) is isomorphic to the direct product (£* x SO(2n + 1).Hence BGO(2n + 1) is homotopic to the direct product B(P* x BSO(2n + 1), with cohomology ring the polynomial ring ZZ/(2)[\,W2iW3,...,W2 n +i] where for 2 > H*(X) on the cohomology ring of X, which is graded of degree -1, with square zero (see Lemma 2.2 below).In terms of the action /J, : W* x X -» X and the projection p : (U* x X -► X, it is given by the formula JJ,* -p* = 77 Y over some base F, recall that we have a long exact Gysin sequence ... 4 H^Y) ^ H^X) 4 fr*-1 ^) 4 ir +1 (y) 4 • ■ • We show (Lemma 2.3) that in this case, the derivation s on H*(X) equals the composite s = TT* o d As 0(2n) C GO(2n) is normal with quotient (E*, BO(2n) can be regarded as the total space of a principal (E*-bundle BO(2n) -> BGO{2n).The resulting action of (E* on BO(2n) gives a derivation s on the ring H*(BO(2n)) = ^/(2)[i(;i,...,u^n] w i^ /x* -p* -r; 0 5.The last equality enables us to write the following expression for 5 (see Lemma 3.4 below) = £w2i i=l