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[parent] antipodal map on $S^n$ is homotopic to the identity if and only if $n$ is odd (Derivation)
Lemma   If $ X\colon S^n\to S^n$ is a unit vector field, then there is a homotopy between the antipodal map on $ S^{n}$ and the identity map.
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 $ \Vert H(v,t)\Vert=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. $ \qedsymbol$
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 $ \Vert X(v)\Vert=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. $ \qedsymbol$

Bibliography

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 $S^n$ is homotopic to the identity if and only if $n$ 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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