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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 $S^n$ as a subspace of $R^{n+1}$ and define $H\colon S^n\times[0,1]\to R^{n+1}$ by $H(v,t)=(\cos\pi t)v+(\sin\pi t)X(v)$ Since $X$ is a unit vector field, $X(v)\perp v$ for any $v\in S^n$ Hence $\|H(v,t)\|=1$ so $H$ is into $S^n$ Finally observe that $H(v,0)=v$ and $H(v,1)=-v$ Thus $H$ is a homotopy between the antipodal map and the identity map. 
Proposition The antipodal map $A\colon S^n\to S^n$ is homotopic to the identity if and only if $n$ is odd.
Proof. If $n$ is even, then the antipodal map $A$ is the composition of an odd number of reflections. It therefore has degree $-1$ Since the degree of the identity map is $+1$ the two maps are not homotopic.
Now suppose $n$ is odd, say $n=2k-1$ Regard $S^n$ has a subspace of $\mathbb{R}^{2k}$ So each point of $S^n$ has coordinates $(x_1,\dots,x_{2k})$ with $\sum_i x_i^2=1$ Define a map $X\colon\mathbb{R}^{2k}\to\mathbb{R}^{2k}$ by $X(x_1,x_2,\dots,x_{2k-1},x_{2k})=(-x_2,x_1,\dots,-x_{2k},x_{2k-1})$ pairwise swapping coordinates and negating the even coordinates. By construction, for any $v\in S^n$ we have that $\|X(v)\|=1$ and $X(v)\perp v$ Hence $X$ is a
unit vector field. Applying the lemma, we conclude that the antipodal map is homotopic to the identity. 
- 1
- Hatcher, A. Algebraic topology, Cambridge University Press, 2002.
- 2
- Munkres, J. Elements of algebraic topology, Addison-Wesley, 1984.
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Cross-references: coordinates, point, maps, degree, reflections, composition, even, odd, identity, homotopic, subspace, identity map, antipodal map, homotopy, field, unit vector
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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 3072 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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