Show that every map $S^n \to S^n$ can be homotoped to have a fixed point if $n > 0$.

Question

This is exercise 2.2.6 on Hatcher. I have seen same question on MSE but I'm having a hard time visualizing the homotopy that many people described. Wondering if someone can explain to me more explicitly and intuitively.

For example, here in this solution, I don't understand what the "half rotations" mean and could not verify that it is indeed a homotopy: homotopic maps from the sphere to the sphere.

Thanks!

Answer 1

A half rotation is another term for a relfection (since the composition of two reflections is a rotation).

The second answer has a fairly intuitive explanation. In odd dimension we have a homotopy from the antipodal map to the identity, which is a composition of rotations by $\pi$ in the $x_1x_2,\ x_3x_4,\ \cdots,\ x_{2k-1}x_{2k}$ planes.

In even dimension, the "equator" $\{(x_1,\ldots,x_{n-1},0)\in S^n\}$ is an odd dimensional sphere. So just use the above homotopy and ignore the last coordinate.