Submersions of closed manifolds 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.
 A: Phillips theorem is plainly wrong in the compact case, and for fairly non subtle reasons.
Take a closed oriented $3$-manifold $M$. There are plenty of bundle epimorphisms $TM \to \mathbb{R}$ because $M$ is parallelizable; any form without zeroes gives you one. Now very often the kernel is a trivial vector bundle as well, so all possible characteristic classes I can think of are null. However, $f$ is never homotopic to a submersion since $\mathbb{R}$ is not compact.
The truth is that the fact that the immersion theorem holds for closed manifolds is an accident. Other cases of Gromov's h-principle (metric with signed curvature, symplectic structures) show that the h-principle fails completely in the closed case. This is not to say that there are no h-principles for closed manifolds, but definitely not for submersions.
A: Let $F$ be the homotopy fibre of $f$ (ie the space of pairs $(x,u)$, where $x\in M$ and $u$ is a path from $f(x)$ to a specified baspoint in $N$).  If $f$ is homotopic to a submersion $f'$, then (using Ehresmann) $F$ will be homotopy equivalent to a closed manifold $(f')^{-1}\{\text{point}\}$, and in particular, it will be equivalent to a finite CW complex.  I do not know whether the converse is also true, but I suspect that any counterexamples would be quite exotic.  So the first thing to do is to try to estimate the size of $F$.
Let $k$ be a field, and take cohomology with coefficients in $k$.  At least is $N$ is simply connected, there is an Eilenberg-Moore spectral sequence $\text{Tor}_{H^*N}(H^*M,k)\Rightarrow H^*F$, where $F$ is the homotopy fibre.  If $H^*M$ is a free module over $H^*N$ then this collapses to an isomorphism $H^*F=H^*M\otimes_{H^*N}k$, and this will be a finite-dimensional $k$-algebra.  I think it will also automatically have Poincare duality (provided that $M$ and $N$ are oriented).  If $H^*M$ is not free over $H^*N$ then the $E_2$ page will typically be very large, and in some cases it may be possible to show that no possible pattern of differentials will leave a finite-dimensional $E_\infty$ page.  If so, we can conclude that $f$ is not homotopic to a submersion.
Alternatively, if $f$ is homotopic to a submersion then the tangent bundle $\tau_M$ maps surjectively to $f^*(\tau_N)$, which implies that the Stiefel-Whitney polynomial $w(\tau_M,t)=\sum_iw_i(\tau_M)t^{\dim(M)-i}$ is divisible by $f^*w(\tau_N,t)$ in $H^*(M;\mathbb{Z}/2)[t]$.  This should generally be easy to check.  There is a similar criterion with Chern polynomials of the complexified tangent bundles in integral cohomology. 
