I'm looking for examples of "fake" projective spaces.
Question: Are there smooth manifolds other than $\mathbb{C}\mathbb{P}^n$ whose cohomology ring is the truncated polynomial ring $\mathbb{K}[h]/h^{n+1}$ with $h$ of degree 2?
I know that there is a classification of so-called fake projective planes in the world of algebraic geometry and results restricting the existence of higher-dimensional examples.
However, I'm interested more in the topological question so don't need these manifolds to arise as varieties, but I do want to have the same ring structure. I don't mind about what coefficient ring we work over.
Has anyone come across examples of such things?
Edit Sorry, I should clarify: I guess I'm really looking for something with the same cohomology ring as $\mathbb{C}\mathbb{P}^n$ but, say, nontrivial fundamental group. Thanks for those answers about exotic smooth structures though.
