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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.

edited Oct 31 2011 at 19:06

asked Oct 31 2011 at 13:57

Richard
51●2

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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 2011 at 20:28

YangMills · Answer 1 · 2011-10-31 14:19:31Z UTC

Quote from this paper of Libgober and Wood, page 1.

"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 $n>3$)"

Mark Grant · Answer 2 · 2011-10-31 14:14:56Z UTC

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The Manifold Atlas page

http://www.map.him.uni-bonn.de/Fake_complex_projective_spaces

has plenty of useful information and references.

answered Oct 31 2011 at 14:14

Mark Grant
9,551●1●14●39

Tara Holm · Answer 3 · 2011-10-31 14:11:16Z UTC

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):

http://www.jstor.org/stable/1996202

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 $\mathbb{C}P^n$.

http://www.map.him.uni-bonn.de/Petrie_conjecture

Fake projective spaces

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