The degree of antipodal map. I am trying to solve the problem A map without fixed points - two wrong approaches. But I am not certain about the degree of antipodal map. I my thought, since the preimage of a point $y \in S^k$ is just $-y$, the degree is just $+1$ or $−1$, depending on the orientation of $-y$?

Hint:

The degree of a reflection through a hyperplane is $-1$, because it is an orientation reversing diffeomorphism. The antipodal map on $S^k$ can be written as the composition of $k+1$ reflections through hyperplanes.

Now use the following very nice fact about the notion of degree of a continuous map. Given any continuous $f,g:S^k\to S^k$, the degree of the composition of $f$ and $g$ is the product of the degrees. In other words:

$$\deg(f\circ g)=\deg f\cdot\deg g$$