TITLE: Existence of submersions from spheres into spheres
QUESTION [19 upvotes]: I would like to know if there exist submersions $f\colon \mathbb{S}^4\to \mathbb{S}^2$ and $g\colon\mathbb{S}^6\to \mathbb{S}^2.$ It is simply a question of curiosity.
Any suggestion or ideas are appreciated.

REPLY [17 votes]: In fact there are no submersions from an even sphere $S^{2n}$ into any smooth manifold $M$ of strictly smaller dimension. 
The pullback of the tangent bundle of $M$ along such a submersion would be a nontrivial quotient bundle of the tangent bundle of $S^{2n}$, but the tangent bundle of $S^{2n}$ has no nontrivial quotient bundles. This is because its Euler class is nonzero but the Euler class of any orientable bundle of dimension smaller than $2n$ on $S^{2n}$ is necessarily zero, since it lives in a cohomology group which vanishes. (And every bundle on $S^{2n}$ is orientable.)
This argument doesn't apply in the odd case, where sometimes the tangent bundle is even trivializable, and in fact in the odd case there are smooth fiber bundles
$$S^1 \to S^{2n+1} \to \mathbb{CP}^n$$
for all $n$, as well as smooth fiber bundles
$$S^3 \to S^{4n+3} \to \mathbb{HP}^n$$
for all $n$. Then there is an exceptional smooth fiber bundle
$$S^7 \to S^{15} \to \mathbb{OP}^1 \cong S^8$$
but this doesn't generalize. 
Edit #2: Here's a different strategy, closer to the one employed in Mike's answer, that handles the odd case. Below all cohomology is with integer coefficients.
Suppose that $S^{n+k} \to S^k$ is a submersion, $n \ge 1$. As in Mike's answer, Ehresmann's theorem shows that such a submersion gives rise to a smooth fiber bundle $F \to S^{n+k} \to S^k$ where $F$ is a closed smooth manifold of dimension $n$. If $k = 1$, the long exact sequence in homotopy shows that $\pi_0(F) \cong \mathbb{Z}$, but this contradicts $F$ compact.
Now suppose $k \ge 2$. By taking homotopy fibers twice, such a smooth fiber bundle extends to a fiber sequence
$$\Omega S^{n+k} \to \Omega S^k \to F \to S^{n+k} \to S^k$$
showing that there is a map $\Omega S^k \to F$ whose homotopy fiber is $n+k-2$-connected, hence which induces an isomorphism on cohomology in degrees $i \le n+k-2$. The ring structure on $H^{\bullet}(\Omega S^k)$ is known (via the cohomological Serre spectral sequence): 

if $k$ is odd, it's a divided power algebra on a generator $a$ of degree $k-1$.
if $k$ is even, it's the tensor product of an exterior algebra on a generator $a$ of degree $k-1$ and a divided power algebra on a generator $b$ of degree $2k-2$. 

Since $n+k-2 \ge n$, this determines $H^{\bullet}(F)$ as a ring. 
Now, $F$ is frameable, and in particular is orientable, so we must have $H^n(F) \cong \mathbb{Z}$. Since $H^{\bullet}(\Omega S^k)$ is concentrated in degrees that are a multiple of $k-1$, it follows that $n = d(k-1)$ for some $d$. 
If either $k$ is odd and $d \ge 2$ or $k$ is even and $d \ge 4$, then the ring structure above is incompatible with Poincaré duality due to the divided power structure. This is a bit annoying to spell out because there are cases depending on the parity of both $d$ and $k$, but for example if $k$ is odd then the product of a generator of $H^{k-1}(F)$ and a generator of $H^{(d-1)(k-1)}(F)$ fails to be a generator of $H^n(F)$. 
So either $k$ is odd and $d = 1$, hence $n = k-1$, or $k$ is even and $d = 1, 2, 3$. If $d = 2$ then $F$ has Euler characteristic $1$, and this contradicts that the base $S^k$ has Euler characteristic $2$ but the total space $S^{3k-3}$ has Euler characteristic $0$; it also contradicts $F$ frameable. The conclusion is that 

If there is a submersion $S^{n+k} \to S^k, n \ge 1$, then either $k \ge 3$ is odd and $n = k-1$ or $k \ge 2$ is even and $n = k-1$ or $n = 3k-3$.

When $k = 2$ this gives that either $n = 1$, which we already know occurs, or $n = 3$, which is ruled out in Mike's answer. 
When $k \ge 3$ and $n = k-1$, the long exact sequence in homotopy shows that if $F \to S^{2k-1} \to S^k$ is a fiber sequence with $k \ge 3$, then $F$ is simply connected. By the computation above, $F$ is an integral homology sphere, and a simply connected integral homology sphere must be homotopy equivalent to a sphere, and so by the topological Poincaré conjecture must be homeomorphic to a sphere. Adams classified all fibrations where all three spaces are spheres, and barring the degenerate case $S^0 \to S^1 \to S^1$ these are the only possibilities for the dimensions of the spheres: 
$$S^1 \to S^3 \to S^2$$
$$S^3 \to S^7 \to S^4$$
$$S^7 \to S^{15} \to S^8.$$
The only remaining case is that $k$ is even and $n = 3k-3$, which I asked a question about here.