The complex cobordism of BSOn

Abstract

In this article, we compute ${\mathit{\text{MU}}}^{*}\left(\mathit{\text{BSO}}\right(2m\left)\right)$ and show that it is generated as an ${\mathit{\text{MU}}}^{*}$-algebra by Conner-Floyd Chern classes $c_i$ and one $2m$-dimensional element $y_m$. The case $\mathit{\text{BO}}\left(n\right)$ was studied by W. S. Wilson, and the case $\mathit{\text{BSO}}(2m+1)$ is derived directly from the result. We obtain the result for $\mathit{\text{BSO}}\left(2m\right)$ by using (equivariant) stratification methods introduced to compute Chow rings by Guillot, Molina, Vezzosi, and Vistoli.