I am studying a homogeneous space and would like to know its cohomology groups. Using some sequences and fibrations I have figured out some of these groups, but largely in terms of the cohomology group of another space. That space is the complex Grassmannian:
The cohomology ring of the Grassmannian is complicated. The treatment in http://www-personal.umich.edu/~jblasiak/grassmannian.pdf is understandable and I am working through it. However, I am a little bit confused about how to actually recover the "classical" cohomology groups from something like example 5.4, page 15 in the pdf. I find myself in the strange position of being told that these cohomology groups are known and, although complicated, possible to write down, without seeing how.