Borsuk–Ulam theorem

In mathematics, the Borsuk–Ulam theorem (BUT), 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 $f: S^n \to R^n$ is continuous then there exists an $x\in S^n$ such that: $f(-x)=f(x)$ .

The case $n=1$ 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.

The case $n=2$ 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.

BUT has several equivalent statements in terms of odd functions. Recall that $S^n$ is the n-sphere and $B^n$ is the n-ball:

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

History

According to Matoušek (2003, p. 25), the first historical mention of the statement of BUT appears in Lyusternik & Shnirel'man (1930). The first proof was given by Karol Borsuk (1933), where the formulation of the problem was attributed to Stanislaw Ulam. Since then, many alternative proofs have been found by various authors, as collected by Steinlein (1985).

Equivalent statements

The following statements are equivalent to BUT.^[1]

With odd functions

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

BUT is equivalent to the following statement: A continuous odd function from an n-sphere into Euclidean n-space has a zero. PROOF:

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

With retractions

Define a retraction as a function $h: S^n \to S^{n-1}$ .

BUT is equivalent to the following claim: there is no continuous odd retraction.

PROOF: If BUT is correct, then every continuous odd function from $S^n$ must include 0 in its range. However, $0 \notin S^{n-1}$ so there cannot be a continuous odd function whose range is $S^{n-1}$ .

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

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

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

Proofs

1-dimensional case

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

Let $g$ be an odd real-valued function on a circle. Pick an arbitrary $x$ . If $g(x)=0$ then we are done. Otherwise, w.l.o.g. $g(x)>0$ . But $g(-x)<0$ . Hence, by the IVT there is a point $y$ between $x$ and $-x$ on which $g(y)=0$ .

General case - algebraic topology proof

BUT can be proved by the stronger statement that every odd function $h^*: S^{n-1} \to S^{n-1}$ has odd degree.

Assume by contradiction that there exists a continuous odd retraction $h: S^n \to S^{n-1}$ .

Let $h^*: S^{n-1} \to S^{n-1}$ be the restriction of $h$ to the equator. $h^*$ is also odd and thus has non-zero degree.

By construction, $h^*$ extends to the whole upper hemisphere of $S^n$ , and as such is null-homotopic . A null-homotopic map has degree zero, contradicting our only assumption, namely that $h$ exists.

General case - combinatorial proof

BUT can be proved from Tucker's lemma. ^[1] ^[2]^[3]

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

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

The absolute value of the label is the index of the coordinate with the highest absolute value of g: $|l(v)| = \arg\max_k (g(v)_k)$ .
The sign of the label is the sign of g, so that: $l(v) = \sgn (g(v)) |l(v)|$ .

Because g is odd, the labeling is also odd: $l(-v) = -l(v)$ . Hence, by Tucker's lemma, there are two adjacent vertices $u, v$ with opposite labels. Assume w.l.o.g. that the labels are $l(u)=1, l(v)=-1$ . By definition of l, this means that in both $g(u)$ and $g(v)$ , coordinate #1 is the largest coordinate; in $g(u)$ this coordinate is positive while in $g(v)$ it is negative. By the construction of the triangulation, the distance between $g(u)$ and $g(v)$ is at most $\epsilon$ ; this means that both $|g(u)|$ and $|g(v)|$ are bounded by $\epsilon$ .

The above is true for every $\epsilon$ ; hence there must be a point u in which $|g(u)|=0$ .

Corollaries

No subset of Rⁿ is homeomorphic to Sⁿ.
The Ham sandwich theorem: For any compact sets A₁, ..., A_n in Rⁿ we can always find a hyperplane dividing each of them into two subsets of equal measure.

Equivalent results

Above we showed how to prove BUT from Tucker's lemma. The converse is also true: it is possible to prove Tucker's lemma from BUT. 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 can be reduced to the other result in its column.^[4]

Algebraic topology	Combinatorics	Set covering
Brouwer fixed-point theorem	Sperner's lemma	KKM lemma
Borsuk–Ulam theorem	Tucker's lemma	Lusternik–Schnirelmann theorem

Generalizations

1. In the original BUT, 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 Rⁿ containing the origin (Here, symmetric means that if x is in the subset then -x is also in the subset).^[5]

2. Consider the function A which maps a point to its antipodal point: A(x)=-x. Note that A(A(x))=x. The original BUT claims that there is a point x in which f(A(x))=f(x). In general, this is true also for every function A for which A(A(x))=x.^[6] However, in general this is not true for other functions A.^[7]

Notes

1 2 Prescott, Timothy (2002). "Extensions of the Borsuk-Ulam Theorem (Thesis)" (PDF). Harvey Mudd College. Retrieved 25 May 2015.
↑ "A constructive proof of Tuckers combinatorial lemma". Journal of Combinatorial Theory, Series A 30: 321–325. doi:10.1016/0097-3165(81)90027-3.
↑ "Consensus-halving via theorems of Borsuk-Ulam and Tucker". Mathematical Social Sciences 45: 15–25. doi:10.1016/s0165-4896(02)00087-2.
↑ "A Borsuk–Ulam Equivalent that Directly Implies Sperner's Lemma". The American Mathematical Monthly 120 (4): 346. 2013. doi:10.4169/amer.math.monthly.120.04.346.
↑ Hazewinkel, Michiel, ed. (2001), "Borsuk fixed-point theorem", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
↑ "On Theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobo and Dyson, I". The Annals of Mathematics 60: 262. doi:10.2307/1969632.
↑ Jens Reinhold, Faisal, Sergei Ivanov. "Generalization of Borsuk-Ulam". Math Overflow. Retrieved 18 May 2015.

References

Borsuk, K. (1933). "Drei Sätze über die n-dimensionale euklidische Sphäre". Fund. Math. 20: 177–190.
Lyusternik, L.; Shnirel'man, S. (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 3-540-00362-2.
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. doi:10.2307/2975293.

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