While I was able to prove that there is no surjective, smooth map $f:S^1 \to S^1 \times S^1 \times S^1$ but I am unable to prove this.
Suppose such a map exists, then by sard's theorem, it has a regular value. Hence there exists a unique $x \in S^2$ where $d_xf$ is surjective. But I am not able to proceed further. Any hint?