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# antipodal

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.

Defines:

antipodal points, antipodal map

Related:

DiametralPoints

Type of Math Object:

Definition

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Reference

## Mathematics Subject Classification

51M05*no label found*15-00

*no label found*

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new question: Prove a formula is part of the Gentzen System by LadyAnne

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