Borsuk–Ulam theorem

In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.

can be illustrated by saying that there always exist a pair of opposite points on the Earth's equator with the same temperature.

This assumes the temperature varies continuously in space, which is, however, not always the case.

is often illustrated by saying that at any moment, there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures, assuming that both parameters vary continuously in space.

The Borsuk–Ulam theorem has several equivalent statements in terms of odd functions.

is the n-ball: According to Matoušek (2003, p. 25), the first historical mention of the statement of the Borsuk–Ulam theorem appears in Lyusternik & Shnirel'man (1930).

The first proof was given by Karol Borsuk (1933), where the formulation of the problem was attributed to Stanisław Ulam.

Since then, many alternative proofs have been found by various authors, as collected by Steinlein (1985).

is called odd (aka antipodal or antipode-preserving) if for every

The Borsuk–Ulam theorem is equivalent to the following statement: A continuous odd function from an n-sphere into Euclidean n-space has a zero.

PROOF: Define a retraction as a function

The Borsuk–Ulam theorem is equivalent to the following claim: there is no continuous odd retraction.

Proof: If the theorem is correct, then every continuous odd function from

so there cannot be a continuous odd function whose range is

Conversely, if it is incorrect, then there is a continuous odd function

The 1-dimensional case can easily be proved using the intermediate value theorem (IVT).

be the odd real-valued continuous function on a circle defined by

can be handled using basic covering theory).

By passing to orbits under the antipodal action, we then get an induced continuous function

between real projective spaces, which induces an isomorphism on fundamental groups.

By the Hurewicz theorem, the induced ring homomorphism on cohomology with

denotes the field with two elements], sends

[3] One can also show the stronger statement that any odd map

has odd degree and then deduce the theorem from this result.

The Borsuk–Ulam theorem can be proved from Tucker's lemma.

Hence, by Tucker's lemma, there are two adjacent vertices

Above we showed how to prove the Borsuk–Ulam theorem from Tucker's lemma.

The converse is also true: it is possible to prove Tucker's lemma from the Borsuk–Ulam theorem.

Additionally, each result in the top row can be deduced from the one below it in the same column.