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. The antipodal map$A:S^n\to S^n$ is homotopic to the identity map if $n$ is odd [1].
2. The degree of the antipodal map is $(-1)^{n+1}$ .

References

1

V. Guillemin, A. Pollack, Differential topology, Prentice-Hall Inc., 1974.