The Chow Ring of a Classifying Space

For any linear algebraic group G, we define a ring CHBG, the ring of characteristic classes with values in the Chow ring (that is, the ring of algebraic cycles modulo rational equivalence) for principal G-bundles over smooth algebraic varieties. We show that this coincides with the Chow ring of any quotient variety (V −S)/G in a suitable range of dimensions, where V is a representation of G and S is a closed subset such that G acts freely outside S. As a result, computing the Chow ring of BG amounts to the computation of Chow groups for a natural class of algebraic varieties with a lot of torsion in their cohomology. Almost nothing is known about this in general. For G an algebraic group over the complex numbers, there is an obvious ring homomorphism CHBG → H(BG,Z). Less obviously, using the results of [42], this homomorphism factors through the quotient of the complex cobordism ring MUBG by the ideal generated by the elements of negative degree in the coefficient ring MU = Z[x1, x2, . . . ], that is, through the ring MUBG⊗MU∗Z. (For clarity, let us mention that the classifying space of a complex algebraic group is homotopy equivalent to that of its maximal compact subgroup. Moreover, every compact Lie group arises as the maximal compact subgroup of a unique complex reductive group.) The most interesting result of this paper is that in all the examples where we can compute the Chow ring of BG, it maps isomorphically to the topologically defined ring MUBG ⊗MU∗ Z. Namely, this is true for finite abelian groups, the symmetric groups, tori, GL(n,C), Sp(2n,C), O(n), SO(2n+ 1), and SO(4). (The computation of the Chow ring for SO(2n + 1) is the result of discussion between me and Rahul Pandharipande. Pandharipande then proceeded to compute the Chow ring of SO(4), which is noticeably more difficult [31].) We also get various additional information about the Chow rings of the symmetric groups, the group G2, and so on. Unfortunately, the map CHBG → MUBG ⊗MU∗ Z is probably not always an isomorphism, since this would imply in particular that MUBG is concentrated in even degrees. By Ravenel, Wilson, and Yagita, MUBG is concentrated in even degrees if all the Morava K-theories of BG are concentrated in even degrees [33]. The latter statement, for all compact Lie groups G, was a plausible conjecture of