$\phi : S^n \to S^n$ with no fixed point

Question

The question is as follows: "Find a continuous map from $S^1$ to $S^1$ with no fixed points. What about for $n > 1$?"

I want to write $S^n = \{(1, \theta_1, \dots, \theta_n) | 0 \leq \theta_i < 2\pi\}$ and define a continuous map $\phi: (1, \theta_1, \dots, \theta_n) \to (1, \theta_1 + \pi, \dots, \theta_n + \pi)$.

Is this correct?

Answer 1

Embed the sphere in the real vector space like this: $S^n=\{v\in\mathbb R^{n+1}:\|v\|=1\}$.

Now just define the map as $\varphi(v)=-v$ for all $v\in S^n$, that's clearly continuous and has no fixed points.