Basis theorem for the Grassamannian

Question

Does anyone know where I can find a good proof for the basis theorem of the cohomology ring of the Grassmannian, or give me a sketch of the proof? I'm already familiar with basic Schubert Calculus.

Answer 1

In "Characteristic Classes" Milnor and Stashev calculate the $\mathbb{F}_2$ cohomology of $BO(n)$ and $BO$ and the integral cohomology of $BU(n)$ and $BU$. The approach is to give a CW decomposition which is enough for the latter and for the former you use characteristic class argument to argue that all the coboundary maps are trivial.