## Proof of the Borsuk-Ulam Theorem

#### Borsuk-Ulam Theorem

Here is an outline of the proof of the Borsuk-Ulam Theorem;
more details can be found in Section 2.6 of Guillemin and Pollack's
book *Differential Topology*.

As there, we will deal with smooth maps, and make use of standard
results like Sard's theorem.

Remember that Borsuk-Ulam says that any *odd* map
**f** from **S**^{n} to itself has odd degree.
Here **f** is called odd if
it is equivariant with respect to the antipodal map: **f(-s)=-f(s)**.

The proof is by induction on **n**. For **n=1**,
lift the map **f** from **S**^{1} to **S**^{1}
to a map **h** from **R** to **R** with **h(x+1)=h(x)+deg(d)**.
But if **f** is odd, we find that **h(x+1/2)=h(x)+k/2** for some
odd integer **k**, and it then follows that **deg(f)=k** is odd.

For the inductive step, let **k** be the degree of **f**,
and let **g** be the restriction of **f**
to the equator **S**^{n-1}. By Sard's theorem, we can
find a point **a** in the image sphere which is a regular value
for **g** and **f**, meaning that it is not in the image of **g**
and it is achieved exactly **k** times by **f**. After a rotation,
we can assume that **a** is exactly the north pole of the sphere.

Because **f** is odd, **k** can be computed as the number
of preimages in the northern hemisphere of the north or south pole.
Defining **f**^{+} as the restriction of **f** to
the northern hemisphere, composed with projection to the equatorial
plane, we find that **k** is the number of preimages of **0**
under **f**^{+}.

Now neither north nor south pole is in the image of **g**, so
we can retract **g** onto the equatorial sphere
and get an odd map from **S**^{n-1} to itself.
Applying the inductive hypothesis, this map has odd degree.

But **g** is the restriction of **f**^{+}
to the boundary. By a standard lemma in differential topology
its degree is then the number of preimages of
the regular value **0**. (If we restrict **f**^{+}
to small spheres around each preimage, they have winding number one each.)
Thus we see that **k** must be odd.

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