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antipodal map on is homotopic to the identity if and only if is odd
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(Derivation)
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Proof. Regard  as a subspace of  and define
![$ H\colon S^n\times[0,1]\to R^{n+1}$ $ H\colon S^n\times[0,1]\to R^{n+1}$](http://images.planetmath.org:8080/cache/objects/7753/l2h/img6.png) by
 . Since  is a unit vector field,
 for any  . Hence
 , so  is into  . Finally observe that  and  . Thus  is a homotopy between the antipodal map and the identity map. 
- 1
- Hatcher, A. Algebraic topology, Cambridge University Press, 2002.
- 2
- Munkres, J. Elements of algebraic topology, Addison-Wesley, 1984.
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"antipodal map on is homotopic to the identity if and only if is odd" is owned by mps.
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(view preamble)
Cross-references: coordinates, point, maps, degree, reflections, composition, even, odd, identity, homotopic, subspace, identity map, antipodal map, homotopy, field, unit vector
There is 1 reference to this entry.
This is version 2 of antipodal map on is homotopic to the identity if and only if is odd, born on 2006-03-21, modified 2006-03-21.
Object id is 7753, canonical name is AntipodalMapOnSnIsHomotopicToTheIdentityIfAndOnlyIfNIsOdd.
Accessed 1933 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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Pending Errata and Addenda
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