antipodal

DefinitionSuppose $x$ and $y$ are points on the$n$-sphere (http://planetmath.org/Sphere) $S^n$.If $x=-y$ then $x$ and$y$ are called antipodal points. Theantipodal map is the map $A:S^n\to S^n$ defined as$A(x)=-x$.

Properties

1. The antipodal map$A:S^n\to S^n$ ishomotopic (http://planetmath.org/HomotopyOfMaps)to the identity map if $n$ is odd [1].

2. The degree (http://planetmath.org/DegreeMapOfSpheres) of theantipodal map is ${(-1)}^{n+1}$.