A continuous map is odd (or antipodal) if for all .
If then it is easy to produce examples of odd continuous maps . Most obviously, we can just define
If is odd, then we have a well-defined map given by .
Suppose that is continuous and odd. Then the induced map is the identity.
As is odd we have a well-defined map given by . This satisfies , so the right-hand square below commutes:
We claim that the left hand square commutes as well. To see this, define by , so and . Consider a continuous map . Choose a lift . The other lift is then , so , so
Here and are the two lifts of the map , so we see that . This shows that , so the left hand square commutes as claimed.
As the diagram commutes, we see that the maps and are compatible with the maps in the exact sequence obtained in Lemma 23.14. In particular, we have commutative squares
for . It is clear that gives the identity on , and we have seen that all of the maps are isomorphisms, so it follows inductively that gives the identity on for . We also have a commutative square
We have seen that is an isomorphism and is the identity so is also the identity, as claimed. ∎
If , then there are no odd continuous maps from to .
Suppose that is odd and continuous. Define by
It is clear that is odd, so the composite is odd, so the induced map must be the identity by Proposition 24.4. In particular, is nonzero.
On the other hand, we can define by
(Using the fact that and are orthogonal, we see that this does indeed lie in .) This gives a homotopy between and a constant map, which implies that is also homotopic to a constant map, so . This contradiction shows that no such map can exist. ∎
Let be a continuous map, with . Then there is a point with .
Suppose (for a contradiction) that no such point exists, so is always nonzero. We can then define by . It is easy to check that this is continuous and antipodal, which contradicts Theorem 24.5, as required. ∎
There are two opposite points on the Earth’s surface that have the same temperature and also the same atmospheric pressure, as we see by considering the map given by
Let , and be three reasonable subsets of . Then there is a plane such that for each set , half of the volume lies on one side of , and half of the volume lies on the other side.
We could have a sandwich, with the top slice of bread filling the set , and cheese filling the set , and the bottom slice of bread filling . The theorem then says that we can make a single straight cut with a knife to share all three components equally.
Suppose that the sets are solid balls, with centres in general position. Then there is a unique possible choice for , namely the plane passing through , and .
We will not be very rigorous about what “reasonable” means, but we will make some comments here and in the body of the proof. To start with, each set should be bounded (which implies that the volume is finite) and the volume should also not be zero.
For any unit vector and we put , so . We then put
so is a plane and is the half-space on one side of that plane. Note that the plane is the same as , and is the half-space on the opposite side to .
We now define by
It is not too hard to check that this is continuous. By Corollary 24.6, there exists with , which means that for . In other words, the plane bisects each of the sets . ∎