The Weakly Complex Bordism of Lie Groups

1. Preliminaries. Let X be the class of compact 1 connected semisimple Lie groups; 3C'C3C is the following set of groups, Sp(w), SU(w), Spin(w), G2, F*, EQ, E7, Es, U*(X) the weakly complex bordism of X [ l ] and A the ring U*(pt) = Z[Fi , F2, • • • ]. A is the weakly complex bordism ring defined by Milnor. The generators F,are weakly complex manifolds of dim 2i. The bordism class of a weakly complex manifold M is determined by its Milnor numbers [2] sw[ikf ] for co ranging over all partitions of n. In particular, the generators F,can be chosen so that s»(F*) = 1 unless i = p — l for some prime p and in this case Si(Yi)=p; moreover, we assume generators F»chosen so that its Todd genera are 1. I t is possible and convenient to introduce bordism theories with other coefficient rings than A. If T is such a ring, [/*( , T) will denote the resulting theory. Briefly here are some examples: Ap = ZP[YU F2, • • • ], A[1 /F„_ i ]= direct lim 1 / F ^ A and Ap[l/F3 ,_i] = directum l / F ^ A ^ . 1 Let Af = {Mn} denote the stable object of Milnor [ l ] and Zp = SUpE 2 the space obtained by attaching E to S via a map of degree p. Ml%.2 denotes the space of base point preserving maps from Zp to Afn+2. Then Uk(X, Ap) = direct lim ILn+k(X + /\M%+2) X is the disjoint union of X and a point x0U*(X, Ap) is the resulting theory. U^XMUY^i]) = U*(X)® ^ [ l / F p i ] and U*(X, A,[ l /Fpi ] ) U*(X, Ap)®ApA,[l/Fp-.i]. To KC.& there is associated a "generating variety" K8 introduced by Bott [4]. Essentially K8 is the homogeneous space K/K 8 where K is the centralizer of a 1-dimensional torus S^^QK. The dimension of the center of K is 1. The commutator map