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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?

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

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