The anti-podal map and its properties:

Let us now calculate the degree of the anti-podal map $A:S^n\longrightarrow S^n$ given by $A({\bf x}) = -{\bf x}$. The anti-podal map is the composite of reflections in the coordinate hyperplanes and so it suffices to compute the degree of one of them say

$$R_n:(x_1, x_2, \dots, x_{n+1})\mapsto (-x_1, x_2, \dots,x_n, x_{n+1}). \eqno(36.4)$$