antipodal (original) (raw)

DefinitionSuppose x and y are points on then-sphere (http://planetmath.org/Sphere) Sn. If x=-y then x andy are called antipodal points. Theantipodal map is the map A:Sn→Sn defined asA⁢(x)=-x.

Properties

1. 1. The antipodal mapA:Sn→Sn is homotopic (http://planetmath.org/HomotopyOfMaps) to the identity map if n is odd [1].
2. 1. The degree (http://planetmath.org/DegreeMapOfSpheres) of the antipodal map is (-1)n+1.

References

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