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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.

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You can make $\pi_1$ non-trivial by a fairly cheap trick: Take any finitely presented group $G$ which has vanishing $H_1$ and $H_2$ (there are infinitely many of these). Realise $G$ as $\pi_1$ of a homology sphere; this can be done in dimensions $>5$ by a theorem of Haefliger, IIRC. Connect sum the homology sphere with $\mathbb{C} P^n$. –  Tim Perutz Oct 31 '11 at 20:28

3 Answers 3

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 $n>3$)."

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While the same statement for $n=2$ is a wide open conjecture. :) –  Marco Golla Oct 31 '11 at 15:15

The Manifold Atlas page


has plenty of useful information and references.

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Hi Diarmuid, thanks for fixing the link. –  Mark Grant Apr 3 at 12:34

A quick google search on "homotopy projective space" yields results that should interest you. A 1973 paper of Bruce Conrad gives constructions of such spaces (starting with a given one): On manifolds with the homotopy type of complex projective space, Trans. Amer. Math. Soc. 176 (1973), 165-180.

Also related is Petrie's Conjecture. He conjectured that if there is a smooth (non-trivial) circle action on a homotopy projective space $M\simeq \mathbb{C}P^n$, then the Pontryagin class of $M$ must be that of $\mathbb{C}P^n$. Some more information is available at http://www.map.mpim-bonn.mpg.de/Petrie_conjecture.

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