Submersions from compact flat manifold Let $M=\mathbb{R}^n/G$ be a closed flat manifold, and let $F\to M \to N$ be a locally trivial submersion, where $F$ and $N$ are closed manifolds.
My question is simple: are $F$ and $N$ homeomorphic to flat manifolds?
This question seems quite natural to me, and I would expect the answer to this fact to be well known.
Any reference will be welcome.
Thank you in advance.
 A: So, the question was answered by Alexander Lytchak. I am writing down the answer for the sake of completeness.
Consider the fibration between the universal covers $F'\to\tilde{M}\to \tilde{N}$. $\tilde{M}$ is contractible and $\tilde{N}$ is simply connected, thus we can apply the Serre spectral sequence with integral coefficients, and from it we obtain that $F'$ and $\tilde{N}$ are contractible. In fact, if $H^*(F')$ has cohomological dimension $a$, and $H^*(\tilde{N})$ has cohomological dimension $b$, then $H^*(\tilde{M})$ would have cohomological dimension $a+b$ and this has to be $0$.
Then $N$ is acyclic, and from the exact sequence in homotopy so is $F$. Moreover we have and exact sequence
$$ 1\to\pi_1(F)\to \pi_1(M)\to \pi_1(N)\to 1$$
where $\pi_1(M)$ is a Bieberbach group, i.e. a torsion free group with a finite index normal abelian subgroup. As for $\pi_1(F)$, a subgroup of a Bieberbach group is again a Bieberbach group and therefore $F$ is homeomorphic to a flat manifold.
As for $N$, one can prove that there exists a normal abelian subgroup of $\pi_1(N)$ of finite index. Moreover, $\pi_1(N)$ is torsion free, since otherwise there would be a finite cyclic subgroup acting on the contractible manifold $\tilde{N}$ without fixed points. This is not possible, and therefore $\pi_1(N)$ is a Bieberbach group. Again, this implies that $N$ is a Bieberbach manifold.
A: If the submersion is Riemannian, the answer is yes for $N$: Any Riemannian submersion with complete flat total space and compact base must have a flat base space. This was proved by Luis Guijarro  and Peter Petersen in Annales Scientifiques de l’École Normale Supérieure
Volume 30, Issue 5, 1997, Pages 595–603. 
However, this uses more hypotheses than you have (and gets a stronger result than you wish for).
