The Chow ring of the classifying space BSO(2n,C)

الأصل · EN

We compute the Chow ring of the classifying space BSO(2n,) in the sense of Totaro using the fibration Gl(2n)/SO(2n) → BSO(2n) → BGl(2n) and a computation of the Chow ring of Gl(2n)/SO(2n) in a previous paper. We find this Chow ring is generated by Chern classes and a characteristic class defined by Edidin and Graham which maps to 2ⁿ⁻¹ times the Euler class under the usual class map from the Chow ring to ordinary cohomology. Moreover, we show this class represents 1/2ⁿ⁻¹(n-1)! times the nth Chern class of the representation of SO(2n) whose highest weight vector is twice that of the half-spin representation.