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Question. Is there an example of a compact $3$-dimensional Calabi-Yau manifold with finite fundamental group $G$ that does not admit a free action on $S^3$?

This question is motivated by the following: it is known that many simply-connected Clabi-Yau 3-folds admit a singular Lagrangian torus fibration over $S^3$. I don't know if there are exceptions. On the other hand, if $\pi_1$ is finite and we still have a lagrangean torus fibration, one can expect that the base is a lens space. But in this case probably $\pi_1$ of the CY-manifold will be equal to $\pi_1$ of the base.

PS. As Tony Pantev explains, the answer to this question is YES -- there are such examples. On the other hand, if we assume that a finite group $G$ is acting freely on a CY 3-manifold preserving the volume form and preserving a Lagrangian torus fibration, this should impose some very strong restrictions on $G$. I wonder if anyone bothered to work out what is the restriction :).

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up vote 13 down vote accepted

This intuition seems to be only loosely right. There are many smooth compact CY threefolds with large fundamental groups. For instance $\mathbb{Z}/3\times \mathbb{Z}/3$, $\mathbb{Z}/8\times \mathbb{Z}/8$, are allowed fundamental groups and I am pretty sure that those do not act freely on $S^{3}$.

More to the point - the Calabi-Yau threefolds that have these fundamental groups are explicitly constructed and we have a pretty good idea of the shape of (at least one of) their slag torus fibrations. For instance, in the first case the Calabi-Yau fibers by genus one curves over a rational elliptic surface, and the slag fibration is compatible with the genus one fibration. In the second case the Calabi-Yau fibers by abelian surfaces and again the slag fibration is compatible. So guided by the holomorphic picture you can easily imagine a situation where you group acts freely on the CY, preserves the slag torus fibration, and the induced action on the base of the fibration is not free. The only thing you can conclude really is that the action of the group on any fiber sitting over a fixed point in the base is free. This is possible to arrange on a torus by taking action by translations.

So, even if your fundamental group happens to admit some free action on $S^{3}$, this doesn't mean that the action on the base of the slag fibration will be free. And, in general, I don't expect it to be free.

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Tony, thanks a lot for the answer! Would you mind to give a reference for the construction of these manifolds (together with fibrations)? –  Dmitri Aug 17 '11 at 0:25
    
There are many possible constructions of the first one: via toric geometry, via elliptic fibrations, via abelian surface fibrations, etc. The elliptic fibration construction is written for instance in my old paper <a href="arxiv.org/abs/hep-th/0410055">http://arxiv.org/abs/…;. You can see there that the group acts freely on the total space but acts with fixed points on the base of the elliptic fibration. The second Calabi-Yau was originally constructed by Gross-Popescu as a pencil of abelian surfaces with polarizations $(1,8)$. –  Tony Pantev Aug 17 '11 at 0:47
    
Tony thanks a lot for all these references! –  Dmitri Aug 17 '11 at 0:55
    
There is a very nice and detailed description of this CY in the Gross-Pavanelli paper arxiv.org/abs/math/0512182. If you have not seen those, you may also want to take a look at this paper arxiv.org/abs/math/0609728 by Borisov-Hua, and the paper arxiv.org/abs/math/0609728 of Bouchard-Donagi. –  Tony Pantev Aug 17 '11 at 1:52

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