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antipodal (Definition)

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.



"antipodal" is owned by mathcam. [ full author list (2) | owner history (1) ]
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Also defines:  antipodal points, antipodal map

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antipodal map on $S^n$ is homotopic to the identity if and only if $n$ is odd (Derivation) by mps
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Cross-references: odd, identity map, properties, map, points
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This is version 3 of antipodal, born on 2003-09-13, modified 2006-01-14.
Object id is 4731, canonical name is Antipodal.
Accessed 7489 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
 51M05 (Geometry :: Real and complex geometry :: Euclidean geometries and generalizations)
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