Continuous surjection $S^n\to S^m$, $n<m$ [closed]

Question

I had an exam last week on algebraic topology. "Suppose $f:S^n\to S^m$ is continuous, where $n<m$. Prove that $f$ is homotopic to a constant mapping. The fact that $S^n$ minus one point is homeomorphic to $R^n$ is given."

I was running out of time and made a bold statement that $f$ can't be surjection. When I saw that there's construction of continuous surjection $I\to I^2$, I thought that my statement is probably wrong. Is it or not? The above result is easy to prove if $f$ can't be surjection.

Answer 1

$S^1$ can be projected on $I$. and there is a continuous surjection from $I^2$ to $S^2$ : for example, start with $I^2$ glue together two opposite edge, you get a cylinder. Then collapse on two points the other edges (which were now circles) and you get a sphere. Hence the surjection from $I$ to $I^2$ allow to construct a surjection from $S^1$ to $S^2$.