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 $A(x)=-x$.

Properties

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 ${(-1)}^{n+1}$.

Title	antipodal
Canonical name	Antipodal
Date of creation	2013-03-22 13:57:45
Last modified on	2013-03-22 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 15-00
Related topic	DiametralPoints
Defines	antipodal points
Defines	antipodal map