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.

Formally: if ${\displaystyle f:S^{n}\to \mathbb {R} ^{n}}$ is continuous then there exists an ${\displaystyle x\in S^{n}}$ such that: ${\displaystyle f(-x)=f(x)}$.

The case ${\displaystyle n=1}$ can be illustrated by saying that there always exist a pair of opposite points on the Earth's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously in space, which is, however, not always the case.^[1]

The case ${\displaystyle n=2}$ 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. Recall that ${\displaystyle S^{n}}$ is the n-sphere and ${\displaystyle B^{n}}$ is the n-ball:

• If ${\displaystyle g:S^{n}\to \mathbb {R} ^{n}}$ is a continuous odd function, then there exists an ${\displaystyle x\in S^{n}}$ such that: ${\displaystyle g(x)=0}$.
• If ${\displaystyle g:B^{n}\to \mathbb {R} ^{n}}$ is a continuous function which is odd on ${\displaystyle S^{n-1}}$ (the boundary of ${\displaystyle B^{n}}$), then there exists an ${\displaystyle x\in B^{n}}$ such that: ${\displaystyle g(x)=0}$.

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).

The following statements are equivalent to the Borsuk–Ulam theorem.^[2]

A function ${\displaystyle g}$ is called odd (aka antipodal or antipode-preserving) if for every ${\displaystyle x}$: ${\displaystyle g(-x)=-g(x)}$.

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:

• If the theorem is correct, then it is specifically correct for odd functions, and for an odd function, ${\displaystyle g(-x)=g(x)}$ iff ${\displaystyle g(x)=0}$. Hence every odd continuous function has a zero.
• For every continuous function ${\displaystyle f}$, the following function is continuous and odd: ${\displaystyle g(x)=f(x)-f(-x)}$. If every odd continuous function has a zero, then ${\displaystyle g}$ has a zero, and therefore, ${\displaystyle f(x)=f(-x)}$. Hence the theorem is correct.

Define a retraction as a function ${\displaystyle h:S^{n}\to S^{n-1}.}$ 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 ${\displaystyle S^{n}}$ must include 0 in its range. However, ${\displaystyle 0\notin S^{n-1}}$ so there cannot be a continuous odd function whose range is ${\displaystyle S^{n-1}}$.

Conversely, if it is incorrect, then there is a continuous odd function ${\displaystyle g:S^{n}\to {\mathbb {R}}^{n}}$ with no zeroes. Then we can construct another odd function ${\displaystyle h:S^{n}\to S^{n-1}}$ by:

${\displaystyle h(x)={\frac {g(x)}{|g(x)|}}}$

since ${\displaystyle g}$ has no zeroes, ${\displaystyle h}$ is well-defined and continuous. Thus we have a continuous odd retraction.

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

Let ${\displaystyle g}$ be the odd real-valued continuous function on a circle defined by ${\displaystyle g(x)=f(x)-f(-x)}$. Pick an arbitrary ${\displaystyle x}$. If ${\displaystyle g(x)=0}$ then we are done. Otherwise, without loss of generality, ${\displaystyle g(x)>0.}$ But ${\displaystyle g(-x)<0.}$ Hence, by the IVT, there is a point ${\displaystyle y}$ between ${\displaystyle x}$ and ${\displaystyle -x}$ at which ${\displaystyle g(y)=0}$.

Assume that ${\displaystyle h:S^{n}\to S^{n-1}}$ is an odd continuous function with ${\displaystyle n>2}$ (the case ${\displaystyle n=1}$ is treated above, the case ${\displaystyle n=2}$ can be handled using basic covering theory). By passing to orbits under the antipodal action, we then get an induced continuous function ${\displaystyle h':\mathbb {RP} ^{n}\to \mathbb {RP} ^{n-1}}$ between real projective spaces, which induces an isomorphism on fundamental groups. By the Hurewicz theorem, the induced ring homomorphism on cohomology with ${\displaystyle \mathbb {F} _{2}}$ coefficients [where ${\displaystyle \mathbb {F} _{2}}$ denotes the field with two elements],

${\displaystyle \mathbb {F} _{2}[a]/a^{n+1}=H^{*}\left(\mathbb {RP} ^{n};\mathbb {F} _{2}\right)\leftarrow H^{*}\left(\mathbb {RP} ^{n-1};\mathbb {F} _{2}\right)=\mathbb {F} _{2}[b]/b^{n},}$

sends ${\displaystyle b}$ to ${\displaystyle a}$. But then we get that ${\displaystyle b^{n}=0}$ is sent to ${\displaystyle a^{n}\neq 0}$, a contradiction.^[3]

One can also show the stronger statement that any odd map ${\displaystyle S^{n-1}\to S^{n-1}}$ has odd degree and then deduce the theorem from this result.

The Borsuk–Ulam theorem can be proved from Tucker's lemma.^[2][4][5]

Let ${\displaystyle g:S^{n}\to \mathbb {R} ^{n}}$ be a continuous odd function. Because g is continuous on a compact domain, it is uniformly continuous. Therefore, for every ${\displaystyle \epsilon >0}$, there is a ${\displaystyle \delta >0}$ such that, for every two points of ${\displaystyle S_{n}}$ which are within ${\displaystyle \delta }$ of each other, their images under g are within ${\displaystyle \epsilon }$ of each other.

Define a triangulation of ${\displaystyle S_{n}}$ with edges of length at most ${\displaystyle \delta }$. Label each vertex ${\displaystyle v}$ of the triangulation with a label ${\displaystyle l(v)\in {\pm 1,\pm 2,\ldots ,\pm n}}$ in the following way:

• The absolute value of the label is the index of the coordinate with the highest absolute value of g: ${\displaystyle |l(v)|=\arg \max _{k}(|g(v)_{k}|)}$.
• The sign of the label is the sign of g, so that: ${\displaystyle l(v)=\operatorname {sgn}(g(v))|l(v)|}$.

Because g is odd, the labeling is also odd: ${\displaystyle l(-v)=-l(v)}$. Hence, by Tucker's lemma, there are two adjacent vertices ${\displaystyle u,v}$ with opposite labels. Assume w.l.o.g. that the labels are ${\displaystyle l(u)=1,l(v)=-1}$. By the definition of l, this means that in both ${\displaystyle g(u)}$ and ${\displaystyle g(v)}$, coordinate #1 is the largest coordinate: in ${\displaystyle g(u)}$ this coordinate is positive while in ${\displaystyle g(v)}$ it is negative. By the construction of the triangulation, the distance between ${\displaystyle g(u)}$ and ${\displaystyle g(v)}$ is at most ${\displaystyle \epsilon }$, so in particular ${\displaystyle |g(u)_{1}-g(v)_{1}|=|g(u)_{1}|+|g(v)_{1}|\leq \epsilon }$ (since ${\displaystyle g(u)_{1}}$ and ${\displaystyle g(v)_{1}}$ have opposite signs) and so ${\displaystyle |g(u)_{1}|\leq \epsilon }$. But since the largest coordinate of ${\displaystyle g(u)}$ is coordinate #1, this means that ${\displaystyle |g(u)_{k}|\leq \epsilon }$ for each ${\displaystyle 1\leq k\leq n}$. So ${\displaystyle |g(u)|\leq c_{n}\epsilon }$, where ${\displaystyle c_{n}}$ is some constant depending on ${\displaystyle n}$ and the norm ${\displaystyle |\cdot |}$ which you have chosen.

The above is true for every ${\displaystyle \epsilon >0}$; since ${\displaystyle S_{n}}$ is compact there must hence be a point u in which ${\displaystyle |g(u)|=0}$.

• No subset of ${\displaystyle \mathbb {R} ^{n}}$ is homeomorphic to ${\displaystyle S^{n}}$
• The ham sandwich theorem: For any compact sets A₁, ..., A_n in ${\displaystyle \mathbb {R} ^{n}}$ we can always find a hyperplane dividing each of them into two subsets of equal measure.

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. Therefore, these two theorems are equivalent. There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in the top row can be deduced from the one below it in the same column.^[6]

• In the original theorem, the domain of the function f is the unit n-sphere (the boundary of the unit n-ball). In general, it is true also when the domain of f is the boundary of any open bounded symmetric subset of ${\displaystyle \mathbb {R} ^{n}}$ containing the origin (Here, symmetric means that if x is in the subset then -x is also in the subset).^[7]
• More generally, if ${\displaystyle M}$ is a compact n-dimensional Riemannian manifold, and ${\displaystyle f:M\rightarrow \mathbb {R} ^{n}}$ is continuous, there exists a pair of points x and y in ${\displaystyle M}$ such that ${\displaystyle f(x)=f(y)}$ and x and y are joined by a geodesic of length ${\displaystyle \delta }$, for any prescribed ${\displaystyle \delta >0}$.^[8][9]
• Consider the function A which maps a point to its antipodal point: ${\displaystyle A(x)=-x.}$ Note that ${\displaystyle A(A(x))=x.}$ The original theorem claims that there is a point x in which ${\displaystyle f(A(x))=f(x).}$ In general, this is true also for every function A for which ${\displaystyle A(A(x))=x.}$^[10] However, in general this is not true for other functions A.^[11]

• Topological combinatorics
• Necklace splitting problem
• Ham sandwich theorem
• Kakutani's theorem (geometry)
• Imre Bárány

• Borsuk, Karol (1933). "Drei Sätze über die n-dimensionale euklidische Sphäre" (PDF). Fundamenta Mathematicae (in German). 20: 177–190. doi:10.4064/fm-20-1-177-190. Archived (PDF) from the original on 2022-10-09.
• Lyusternik, Lazar; Shnirel'man, Lev (1930). "Topological Methods in Variational Problems". Issledowatelskii Institut Matematiki I Mechaniki Pri O. M. G. U. Moscow.
• Matoušek, Jiří (2003). Using the Borsuk–Ulam theorem. Berlin: Springer Verlag. doi:10.1007/978-3-540-76649-0. ISBN 978-3-540-00362-5.
• Steinlein, H. (1985). "Borsuk's antipodal theorem and its generalizations and applications: a survey. Méthodes topologiques en analyse non linéaire". Sém. Math. Supér. Montréal, Sém. Sci. OTAN (NATO Adv. Study Inst.). 95: 166–235.
• Su, Francis Edward (Nov 1997). "Borsuk-Ulam Implies Brouwer: A Direct Construction" (PDF). The American Mathematical Monthly. 104 (9): 855–859. CiteSeerX 10.1.1.142.4935. doi:10.2307/2975293. JSTOR 2975293. Archived from the original (PDF) on 2008-10-13. Retrieved 2006-04-21.

• Who (else) cares about topology? Stolen necklaces and Borsuk-Ulam on YouTube
• The Borsuk-Ulam Explorer. An interactive illustration of Borsuk-Ulam Theorem.