# proof of Borsuk-Ulam theorem

Proof of the Borsuk-Ulam theorem: I’m going to prove a stronger statement than the one given in the statement of the Borsak-Ulam theorem here, which is:

Every odd (that is, antipode-preserving) map $f\mathrm{:}{S}^{n}\mathrm{\to}{S}^{n}$ has odd degree.

Proof: We go by induction^{} on $n$. Consider the pair $({S}^{n},A)$ where $A$ is the equatorial sphere.
$f$ defines a map

$$\stackrel{~}{f}:\mathbb{R}{P}^{n}\to \mathbb{R}{P}^{n}$$ |

. By cellular approximation, this may be
assumed to take the hyperplane at infinity (the $n-1$-cell of the standard cell structure^{} on
$\mathbb{R}{P}^{n}$) to itself. Since whether a map lifts to a covering depends only on its homotopy
class, $f$ is homotopic^{} to an odd map taking $A$ to itself. We may assume that $f$ is such a map.

The map $f$ gives us a morphism^{} of the long exact sequences:

$$\begin{array}{ccccccccc}\hfill {H}_{n}(A;{\mathbb{Z}}_{2})\hfill & \hfill \stackrel{i}{\to}\hfill & \hfill {H}_{n}({S}^{n};{\mathbb{Z}}_{2})\hfill & \hfill \stackrel{j}{\to}\hfill & \hfill {H}_{n}({S}^{n},A;{\mathbb{Z}}_{2})\hfill & \hfill \stackrel{\partial}{\to}\hfill & \hfill {H}_{n-1}(A;{\mathbb{Z}}_{2})\hfill & \hfill \stackrel{i}{\to}\hfill & \hfill {H}_{n-1}({S}^{n},A;{\mathbb{Z}}_{2})\hfill \\ \hfill {f}^{*}\downarrow \hfill & & \hfill {f}^{*}\downarrow \hfill & & \hfill {f}^{*}\downarrow \hfill & & \hfill {f}^{*}\downarrow \hfill & & \hfill {f}^{*}\downarrow \hfill & & \\ \hfill {H}_{n}(A;{\mathbb{Z}}_{2})\hfill & \hfill \stackrel{i}{\to}\hfill & \hfill {H}_{n}({S}^{n};{\mathbb{Z}}_{2})\hfill & \hfill \stackrel{j}{\to}\hfill & \hfill {H}_{n}({S}^{n},A;{\mathbb{Z}}_{2})\hfill & \hfill \stackrel{\partial}{\to}\hfill & \hfill {H}_{n-1}(A;{\mathbb{Z}}_{2})\hfill & \hfill \stackrel{i}{\to}\hfill & \hfill {H}_{n-1}({S}^{n},A;{\mathbb{Z}}_{2})\hfill \end{array}$$ |

Clearly, the map ${f|}_{A}$ is odd, so by the induction hypothesis, ${f|}_{A}$ has odd degree.
Note that a map has odd degree if and only if ${f}^{*}:{H}_{n}({S}^{n};{\mathbb{Z}}_{2})\to {H}_{n}({S}^{n},{\mathbb{Z}}_{2})$ is an
isomorphism^{}. Thus

$${f}^{*}:{H}_{n-1}(A;{\mathbb{Z}}_{2})\to {H}_{n-1}(A;{\mathbb{Z}}_{2})$$ |

is an isomorphism. By the commutativity of the diagram, the map

$${f}^{*}:{H}_{n}({S}^{n},A;{\mathbb{Z}}_{2})\to {H}_{n}({S}^{n},A;{\mathbb{Z}}_{2})$$ |

is
not trivial. I claim it is an isomorphism. ${H}_{n}({S}^{n},A;{\mathbb{Z}}_{2})$ is generated by cycles $[{R}^{+}]$ and
$[{R}^{-}]$ which are the fundamental classes^{} of the upper and lower hemispheres, and the antipodal
map exchanges these. Both of these map to the fundamental class of $A$,
$[A]\in {H}_{n-1}(A;{\mathbb{Z}}_{2})$. By the commutativity of the diagram,
$\partial ({f}^{*}([{R}^{\pm}]))={f}^{*}(\partial ([{R}^{\pm}]))={f}^{*}([A])=[A]$. Thus ${f}^{*}([{R}^{+}])=[{R}^{\pm}]$ and ${f}^{*}([{R}^{-}])=[{R}^{\mp}]$ since $f$ commutes with the antipodal map. Thus ${f}^{*}$ is an isomorphism on
${H}_{n}({S}^{n},A;{\mathbb{Z}}_{2})$. Since ${H}_{n}(A,{\mathbb{Z}}_{2})=0$, by the exactness of the sequence^{} $i:{H}_{n}({S}^{n};{\mathbb{Z}}_{2})\to {H}_{n}({S}^{n},A;{\mathbb{Z}}_{2})$ is injective^{}, and so by the commutativity of the diagram (or equivalently
by the $5$-lemma) ${f}^{*}:{H}_{n}({S}^{n};{\mathbb{Z}}_{2})\to {H}_{n}({S}^{n};{\mathbb{Z}}_{2})$ is an isomorphism. Thus
$f$ has odd degree.

The other statement of the Borsuk-Ulam theorem is:

There is no odd map ${S}^{n}\mathrm{\to}{S}^{n\mathrm{-}\mathrm{1}}$.

Proof: If $f$ where such a map, consider $f$ restricted to the equator $A$ of ${S}^{n}$. This is an odd map from ${S}^{n-1}$ to ${S}^{n-1}$ and thus has odd degree. But the map

$${f}^{*}{H}_{n-1}(A)\to {H}_{n-1}({S}^{n-1})$$ |

factors through ${H}_{n-1}({S}^{n})=0$, and so must be zero. Thus ${f|}_{A}$ has degree 0, a
contradiction^{}.

Title | proof of Borsuk-Ulam theorem |
---|---|

Canonical name | ProofOfBorsukUlamTheorem |

Date of creation | 2013-03-22 13:10:33 |

Last modified on | 2013-03-22 13:10:33 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Proof |

Classification | msc 54C99 |