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.



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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](http://en.wikipedia.org/wiki/Ehresmann%27s_theorem), and since $M$ is compact, it is equivalent to ask whether $f$ is homotopic to the projection of a locally trivial fibration.



Submersions of closed manifolds
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. 


