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It is known that there is unique differantial topological structure on the elliptic curves or K3 surfaces over $\mathbb{C}$. Since we know tons of Hodge diamonds for Calabi-Yau threefolds, we cannot really expect an easy classification of (differantial) topological structure.

What is known about (differantial) topological structures of Calabi-Yau 3-folds? Are they really different from those of Kahler 3-folds? Are there Calabi-Yau 3-folds that are (diffeomorphic or) homeomorphic but not (complex) deformation equivalent?

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2 Answers 2

There is at least one pair of examples of diffeomorphic but not deformation equivalent three-dimensional Calabi-Yau manifold (Ruan-Gross). The example is explained on pages 47-48 of this paper: http://arxiv.org/pdf/math/9806111.pdf

Otherwise it is of course natural to try to distinguish Calabi Yau three folds by their diffo type. Note that in dimension six two smooth compact manifolds that are homeomorphic are necessarily diffeomorphic, so the classifications up to homeo and diffeo are the same. Classification of simply connected 6-manifolds with torsion free homology according to diffeo is given by a theorem of Wall (the essential bit here is the cubic intersection form on $H^2(M^6,\mathbb Z)$). I am not aware of (current) classification work in this direction for Calabi-Yau 3-fold. But I think someone who would like to do this should use computer (the majority of examples of CY 3-folds are an outcome of a certain computer program). And it seems to me that it should be possible in principle to improve the existing algorithm so that it computes not only betti numbers, but also multiplication on $H^*$ and so the type as well.

Concerning the topology of CY 3-folds, on can say at least that the fundamental group of CY manifolds is finite (or, depending on definition of what you call CY manifold, virtually Abelian). At the same time general Kahler manifolds can have very sophisticated fundamental groups.

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Thank you for the nice answer, Dmitri. I was not aware of the Ruan-Gorss example and it is very interesting. I wonder what makes fundamental group of CY3s finite with your definition. Isn't there Oguis-Sakurai's paper about CY3 with infinite fundamental groups? –  K Kim Feb 28 '13 at 20:50
    
Dear Kim, by Bogomolov-Beuaville theorem every CY manifold has a finite cover that is a product of Tori, hyperkahler manifolds and manifolds $M^n$ such that $H^k(M^n,O)=0$ for $k\ne 0,n$. So for some people "proper" CY manifolds are only those that satisfy the last condition: $H^k(M^n,O)=0$ for $k\ne 0,n$. Such manifolds also have the property that the holonomy group of CY metrics on them coincide with $SU(n)$ (and not smaller than this). Such manifolds do have finite fundamental groups. Maybe for Oguis-Sakurai a Kahler manifold is $CY$ iff it has a holomorphic volume form... –  Dmitri Feb 28 '13 at 21:36
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As Dimitri pointed out, differentiable structure of a real 6-mainfold is essentially determined by the cubic intersection form on the second cohomology group (Wall's theorem). For Calabi-Yau 3-folds, there some classical results in this direction, notably by Pelham Wilson. You may want to take a look at this paper by Atsushi Kanazawa and Pelham Wilson. They showed that there exist some relations among Chern classes and cubic intersection form of a Calabi-Yau 3-fold.

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