Let $f\colon\thinspace M\to N$ be a map of closed smooth manifolds, with $\dim M > \dim N$. Recall that a *submersion* is a smooth map whose differential is surjective at every point in the domain.

Can one give conditions which guarantee that $f$ is homotopic to an submersion?

These conditions would necessarily have to be homotopy invariants. I am thinking there may be something in terms of relations amongst characteristic classes.

I am aware of the theorem of Phillips

MR0208611 (34 #8420) Phillips, Anthony Submersions of open manifolds. Topology 6 1967 171–206.

which says roughly that, *when $M$ is open,* $f\colon\thinspace M\to N$ is homotopic to an immersion if and only if the differential $df\colon\thinspace TM\to TN$ is homotopic to a bundle epimorphism. I'm also aware of the subsequent work of Thomas

MR0225332 (37 #926) Thomas, Emery On the existence of immersions and submersions. Trans. Amer. Math. Soc. 132 1968 387–394.

giving applications of Phillips' theorem. However, I couldn't find any more modern references dealing with the case $M$ closed. Are there any, or is there a good heuristic reason why such conditions cannot be given?

**Remark:** By Ehresmann's Theorem, and since $M$ is compact, it is equivalent to ask whether $f$ is homotopic to the projection of a locally trivial fibration.