Product of linear projective subspaces in complex cobordism ($[{\mathbb{CP}}^k,{\nu }_k][{\mathbb{CP}}^m,{\nu }_m]=[{\mathbb{CP}}^{k+m-n},{\nu }_{k+m-n}]$)

The displayed product applies when $k+m\ge n$. Move one projective subspace by a unitary isotopy into transverse position. The intersection is a linear projective subspace of dimension $k+m-n$, with normal complex bundle the sum of the restricted input normal bundles. If $k+m<n$, representatives can be made disjoint and the product is zero.