Let $\mathcal{E}$ be a vector bundle on $\mathbb{RP}^n$ and $\mathbb{P}\mathcal{E}$ the associated real projective space bundle over $\mathbb{RP}^n$. How can we calculate the cohomology ring of $\mathbb{P}\mathcal{E}$ (say over $\mathbb{Z}_2$)?
From algebraic geometry, I know a nice description for the Chow ring of an algebraic projective space bundle over the complex projective space in terms of Chern classes. Is something similar possible in this real context as well?
Best Answer
What I am saying holds for any reasonable space. The cohomology of the total space of a fiber bundle is a module over the cohomology of the base space by pulling back an element and cupping. The Leray-Hirsch theorem is a theorem about what conditions are necessary on a fiber bundle to have this module be free (i.e. to behave like the cohomology of a product).
With respect to $\mathbb{Z}$ coefficients, the conditions are satisfied by any complex projective space bundle coming from a complex vector bundle. With respect to $\mathbb{Z}/2$ coefficients, the same thing holds for real projective space bundles coming from real vector bundles.
In particular, there is a basis for this cohomology. There is a tautological line bundle over the total space, and it is the case that if $x$ denotes the first Stiefel-Whitney class, the elements $1,x,x^2,\dots,x^n$ form a basis, in particular none of these are 0. Here n is the rank of the vector bundle.
This is all written up in the complex case in these lecture notes by Stephan Stolz: https://www3.nd.edu/~stolz/2020S_Math80440/Index_theory_S2020.pdf
So the point is that all the cohomologies will be isomorphic (though Steenrod operations might detect the difference?). As a sanity check, lets think of the case of a vector bundle over a point. The module structure will be trivial, so the claim is that the cohomology of real projective space is generated by one element in each degree which is exactly what the cohomology should be.
For example, this should prove that the Hopf fibre bundle is not the projective space bundle associated to any real vector bundle. Hopefully someone could separately confirm that.