I'm quoting a question from p. 753 of Gromov's recent paper Singularities, Expanders and Topology of Maps:
Does there exist, for every closed oriented $n$-manifold $X_0$, a closed oriented $n$-manifold $X$ that admits a map $X \to X_0$ of positive degree and, at the same time, can be smoothly fibered over some $Y$ with $dim(Y) = n − 2$ ? (Bogomolov’s original question concerns parametrization of complex algebraic manifolds $X_0$ by algebraic manifolds fibred by surfaces.)
I'm wondering where Bogomolov's question came from, and for what cases it is known and why it may be of interest? Gromov doesn't give a reference, and searching for "Bogomolov conjecture" seems to bring up a different conjecture. Maybe an expert can point me quickly to a reference.
Also, if anyone has partial answers to Gromov's question, I would be interested. I think I can prove it for 3-manifolds, and there may be a 4-manifold topologist who might know something about the 4-dimensional case.
Addendum: I'll add my argument in the 3-dimensional case, then ask a question which would answer Gromov's question in dimension 4.
Any orientable 3-manifold is a cover of $S^3$ branched over the figure 8 knot $K$. The 3-fold cyclic cover branched over $K$ is a Euclidean manifold, which has a finite-sheeted cover which is a 3-torus. The preimage of the branched locus is geodesic, so we may find a product fibering of the 3-torus by tori transverse to the preimage of the branched locus. Thus, any covering of this cover will also fiber (by taking the preimage of the fibering). For a 3-manifold, we may take the common branched cover over $K$ with the 3-torus branched cover. This fibers since it covers the 3-torus, and has a non-zero degree (in fact bounded degree) map to the 3-manifold.
As mentioned in an answer to Ramified cover of 4-sphere, any closed orientable PL 4-manifold is a branched cover over $S^4$ with branched locus a surface. One could try to generalize my 3-dimensional argument to this context. The key fact is that there is a cover of $S^3$ branched over the figure 8 knot which has a fibration transverse to the branched locus. So I ask the question: for any surface in $S^4$, is there a cover of $S^4$ branched over the surface which has a fibration by surfaces which are transverse to the preimage of the branched locus? I have no evidence for or against this question, it just seems like a natural generalization of the 3-D case.
One could also ask the analogous question in higher dimensions and in the algebraic category. But one would need to know the corresponding branched covering result first. For projective algebraic varieties, maybe it would be natural to consider branched covers over projective space, analogous to Belyi's theorem.