For which $n$, where $n$ is a positive integer, is any continuous map from $S^n$ to $S^1 \times S^1$ nulhomotopic.
If every continuous map from some $S^n$ to $S^1 \times S^1$ was nulhomotopic, would this not mean that they had the same fundamental group? Since the fundamental group of $S^n$ is trivial for $n\geq 2$, and $\pi_1 (S^1) = \mathbb{Z}$ and $\pi_1 (S^1 \times S^1) = \mathbb{Z} \times \mathbb{Z}$, is there no such positive integer?
Thanks!
Every $n$ bigger than $1,$ which is another way of saying that the higher homotopy groups of $S^1 \times S^1$ are trivial, which is true because the universal covering space is $\mathbb{R}^2,$ which is acyclic.