Null-homotopic Maps from $S^n$ to $S^1$ for $n \gt 1$.

Question

I'm not sure how to answer this one. Is every continuous map $f:S^2 \to S^1$ null-homotopic? If $n > 1$, where $n$ is a natural number, is every continuous map $f:S^n \to S^1$ null-homotopic?

Answer 1

any such $f$ lifts to the universal cover $\mathbb{R}$ where it is clearly nullhomotopic, then project the nullhomotopy back to $S^1$