antipodal
Definition Suppose $x$ 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. The antipodal map is the map $A:{S}^{n}\to {S}^{n}$ defined as $A(x)=x$.

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

2.
The degree (http://planetmath.org/DegreeMapOfSpheres) of the antipodal map is ${(1)}^{n+1}$.
References
 1 V. Guillemin, A. Pollack, Differential topology, PrenticeHall Inc., 1974.
Title  antipodal 

Canonical name  Antipodal 
Date of creation  20130322 13:57:45 
Last modified on  20130322 13:57:45 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  6 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 51M05 
Classification  msc 1500 
Related topic  DiametralPoints 
Defines  antipodal points 
Defines  antipodal map 