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.
Best Answer
Quote from the first page of the article Uniqueness of the complex structure on Kähler manifolds of certain homotopy types by Libgober and Wood (Journal of Differential Geometry 32, 1990, no. 1, 139-154):
"On the other hand it is known that for every $n>2$ the homotopy type of $\mathbb{CP}^n$ supports infinitely many inequivalent differentiable structures distinguished by their Pontryagin classes (see Montgomery and Yang [25] or Wall [30] for $n=3$ and Hsiang [15] for $n>3$)."