# Finite fundamental groups of 3-dimensional Calabi-Yau manifolds

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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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 $S^{3}$.
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.
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