# antipodal map on ${S}^{n}$ is homotopic to the identity if and only if $n$ is odd

###### Lemma.

If $X\mathrm{:}{S}^{n}\mathrm{\to}{S}^{n}$ 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
$H:{S}^{n}\times [0,1]\to {R}^{n+1}$ by
$H(v,t)=(\mathrm{cos}\pi t)v+(\mathrm{sin}\pi t)X(v)$. Since $X$ is a unit
vector field, $X(v)\u27c2v$ for any $v\in {S}^{n}$. Hence
$\parallel H(v,t)\parallel =1$, 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 $A\mathrm{:}{S}^{n}\mathrm{\to}{S}^{n}$ 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 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^{} $({x}_{1},\mathrm{\dots},{x}_{2k})$ with ${\sum}_{i}{x}_{i}^{2}=1$. Define
a map $X:{\mathbb{R}}^{2k}\to {\mathbb{R}}^{2k}$ by
$X({x}_{1},{x}_{2},\mathrm{\dots},{x}_{2k-1},{x}_{2k})=(-{x}_{2},{x}_{1},\mathrm{\dots},-{x}_{2k},{x}_{2k-1})$,
pairwise swapping coordinates and negating the even coordinates.
By construction, for any $v\in {S}^{n}$, we have that $\parallel X(v)\parallel =1$
and $X(v)\u27c2v$. Hence $X$ is a unit vector field. Applying the
lemma, we conclude that the antipodal map is homotopic to the identity.
∎

## References

- 1 Hatcher, A. Algebraic topology, Cambridge University Press, 2002.
- 2 Munkres, J. Elements of algebraic topology, Addison-Wesley, 1984.

Title | antipodal map on ${S}^{n}$ is homotopic to the identity if and only if $n$ is odd |
---|---|

Canonical name | AntipodalMapOnSnIsHomotopicToTheIdentityIfAndOnlyIfNIsOdd |

Date of creation | 2013-03-22 15:47:33 |

Last modified on | 2013-03-22 15:47:33 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 5 |

Author | mps (409) |

Entry type | Derivation |

Classification | msc 51M05 |

Classification | msc 15-00 |