When is there a submersion from a sphere into a sphere?
This question has been asked and answered on MathOverflow. I have replicated the accepted answer by Tom Goodwillie below.
In most cases $\pi_{n+k}(S^k)$ is a finite group, so that the homotopy fiber of any map $S^{n+k}\to S^k$ is rationally equivalent to $\Omega S^k\times S^{n+k}$ and therefore has homology in arbitrarily high dimensions and cannot be a manifold.
The only exceptions with $n>0$ have $n=k-1$.