EggMath: The White/Yolk Theorem
Proof of the Borsuk-Ulam
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
As there, we will deal with smooth maps, and
make use of standard results like Sard's
Remember that Borsuk-Ulam says that any
odd map f from Sn
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
S1 to S1 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
Sn-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
Sn-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.