Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are compact, smooth manifolds.


I am looking for a counterexample for the following statement:


$E \cong \widetilde{B} \times\_G F'$, where $G$ is a finite quotient of $\pi\_1(B)$ acting on a compact smooth manifold $F'$ and $\widetilde{B}$ is the cover of $B$ associated to that quotient. "$\cong$" stands for a) fiberwise diffeomorphic, b) fiberwise homotopy equivalent.


**Related**


This is one aspect of this question [Compute cohomology of flat fiber bundles - does this always work?](https://mathoverflow.net/questions/282220/compute-cohomology-of-flat-fiber-bundles-does-this-always-work), more precisely formulated.



Are there compact flat fiber bundles with "truly" infinite structure group?
Any finitely presented group occurs as the fundamental group of a smooth compact manifold, so the question reduces to whether we can find a finitely presented group $\pi$ acting on a smooth compact manifold $F$ which does not factor through a finite quotient up to homotopy. In turn, it's enough to find an action whose induced action on the homology groups of $F$ doesn't factor through a finite quotient.


An easy example that comes to mind is $\pi = GL\_2(\mathbb{Z}), F = T^2$, since here the action on homology $H\_1(T^2)$ is even faithful. Any infinite subgroup of $GL\_2(\mathbb{Z})$ will do as well, including the copy of $\mathbb{Z}$ generated by any element of infinite order, and so we can get $3$-manifold examples which are $T^2$-bundles over $S^1$ by taking mapping tori. 


