Explicit homotopy that takes antipodal map to a map with fixed point

Question

homotopic maps from the sphere to the sphere

The link above gives a very intuitive way to show that the result in question holds but could someone give me please the explicit homotopy he is using?

Thanks in advance.

Answer 1

The homotopy is $$H(x_1,x_2,\dots,x_n;t) = (r\cos(t\pi+\alpha),r\sin(t\pi+\alpha),-x_3,\dots,-x_n),$$ where $(r,\alpha)$ is the standard polar coordinates of the point $(x_1,x_2)$, i.e, $$x_1=r\cos(\alpha),\, x_2=r\sin(\alpha).$$

It's obvious that $$H(x_1,x_2,\dots, x_n;0) = (x_1,x_2,-x_3,\dots, -x_n),$$ and $$H(x_1,x_2,\dots, x_n;1)= (-x_1,-x_2,-x_3,\dots, -x_n).$$ I leave you to find the fix point(s) of $H(x_1,x_2,\dots,x_n;0)$.