Definition Suppose $x$ and $y$ are points on the $n$-sphere $S^{n}$. If $x=-y$ then $x$ and $y$ are called antipodal points. The antipodal map is the map $A:S^{n}\to S^{n}$ defined as $A(x)=-x$.

Properties

1. 1.

   The antipodal map$A:S^{n}\to S^{n}$ is homotopic to the identity map if $n$ is odd [1].

2. 2.

   The degree of the antipodal map is $(-1)^{{n+1}}$.