Borsuk–Ulam theorem

Antipodal

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 <math>f: S^n \to \R^n</math> is continuous then there exists an <math>x\in S^n</math> such that: <math>f(-x)=f(x)</math>.

The case <math>n=1</math> 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.<ref name=":0">Template:Cite journal</ref>

The case <math>n=2</math> 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 <math>S^n</math> is the n-sphere and <math>B^n</math> is the n-ball:

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

HistoryEdit

According to Template:Harvtxt, the first historical mention of the statement of the Borsuk–Ulam theorem appears in Template:Harvtxt. The first proof was given by Template:Harvs, 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 Template:Harvtxt.

Equivalent statementsEdit

The following statements are equivalent to the Borsuk–Ulam theorem.<ref name=prescott2002>Template:Cite thesis</ref>

With odd functionsEdit

A function <math>g</math> is called odd (aka antipodal or antipode-preserving) if for every <math>x</math>, <math>g(-x)=-g(x)</math>.

The Borsuk–Ulam theorem is equivalent to each of the following statements:

(1) Each continuous odd function <math>S^n\to \R^n</math> has a zero.

(2) There is no continuous odd function <math> S^n \to S^{n-1}</math>.

Here is a proof that the Borsuk-Ulam theorem is equivalent to (1):

(<math>\Longrightarrow</math>) If the theorem is correct, then it is specifically correct for odd functions, and for an odd function, <math>g(-x)=g(x)</math> iff <math>g(x)=0</math>. Hence every odd continuous function has a zero.

(<math>\Longleftarrow </math>) For every continuous function <math>f:S^n\to \R^n</math>, the following function is continuous and odd: <math>g(x)=f(x)-f(-x)</math>. If every odd continuous function has a zero, then <math>g</math> has a zero, and therefore, <math>f(x)=f(-x)</math>.

To prove that (1) and (2) are equivalent, we use the following continuous odd maps:

the obvious inclusion <math> i: S^{n-1}\to \R^n\setminus \{0\} </math>,
and the radial projection map <math> p: \R^n\setminus \{0\} \to S^{n-1} </math> given by <math>x \mapsto \frac{x}{|x|}</math>.

The proof now writes itself.

<math>((1) \Longrightarrow (2))</math> We prove the contrapositive. If there exists a continuous odd function <math>f:S^n\to S^{n-1}</math>, then <math> i\circ f</math> is a continuous odd function <math>S^n\to \R^n\setminus \{0\}</math>.

<math>((1) \Longleftarrow (2)) </math> Again we prove the contrapositive. If there exists a continuous odd function <math>f:S^n\to \R^{n}\setminus\{0\}</math>, then <math> p\circ f</math> is a continuous odd function <math>S^n\to S^{n-1}</math>.

ProofsEdit

1-dimensional caseEdit

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

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

General caseEdit

Algebraic topological proofEdit

Assume that <math>h: S^n \to S^{n-1}</math> is an odd continuous function with <math>n > 2</math> (the case <math>n = 1</math> is treated above, the case <math>n = 2</math> can be handled using basic covering theory). By passing to orbits under the antipodal action, we then get an induced continuous function <math>h': \mathbb{RP}^n \to \mathbb{RP}^{n-1}</math> between real projective spaces, which induces an isomorphism on fundamental groups. By the Hurewicz theorem, the induced ring homomorphism on cohomology with <math>\mathbb F_2</math> coefficients [where <math>\mathbb F_2</math> denotes the field with two elements],

<math> \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},</math>

sends <math>b</math> to <math>a</math>. But then we get that <math>b^n = 0</math> is sent to <math>a^n \neq 0</math>, a contradiction.<ref name=rotman>Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag Template:ISBN (See Chapter 12 for a full exposition.)</ref>

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

Combinatorial proofEdit

The Borsuk–Ulam theorem can be proved from Tucker's lemma.<ref name=prescott2002/><ref name=FreundTodd1982>Template:Cite journal</ref><ref name=SimmonsSu2003>Template:Cite journal</ref>

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

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

The absolute value of the label is the index of the coordinate with the highest absolute value of g: <math>|l(v)| = \arg\max_k (|g(v)_k|)</math>.
The sign of the label is the sign of g at the above coordinate, so that: <math>l(v) = \sgn (g(v)_{|l(v)|}) |l(v)|</math>.

Because g is odd, the labeling is also odd: <math>l(-v) = -l(v)</math>. Hence, by Tucker's lemma, there are two adjacent vertices <math>u, v</math> with opposite labels. Assume w.l.o.g. that the labels are <math>l(u)=1, l(v)=-1</math>. By the definition of l, this means that in both <math>g(u)</math> and <math>g(v)</math>, coordinate #1 is the largest coordinate: in <math>g(u)</math> this coordinate is positive while in <math>g(v)</math> it is negative. By the construction of the triangulation, the distance between <math>g(u)</math> and <math>g(v)</math> is at most <math>\epsilon</math>, so in particular <math>|g(u)_1 - g(v)_1| = |g(u)_1| + |g(v)_1| \leq \epsilon </math> (since <math>g(u)_1</math> and <math>g(v)_1</math> have opposite signs) and so <math>|g(u)_1| \leq \epsilon</math>. But since the largest coordinate of <math>g(u)</math> is coordinate #1, this means that <math>|g(u)_k| \leq \epsilon</math> for each <math>1 \leq k \leq n</math>. So <math>|g(u)| \leq c_n \epsilon</math>, where <math>c_n </math> is some constant depending on <math>n </math> and the norm <math>|\cdot| </math> which you have chosen.

The above is true for every <math>\epsilon > 0</math>; since <math>S_n</math> is compact there must hence be a point u in which <math>|g(u)|=0</math>.

CorollariesEdit

No subset of <math>\R^n</math> is homeomorphic to <math>S^n</math>
The ham sandwich theorem: For any compact sets A₁, ..., A_n in <math>\R^n</math> we can always find a hyperplane dividing each of them into two subsets of equal measure.

Equivalent resultsEdit

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. Template:Analogous fixed-point theorems

GeneralizationsEdit

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 <math>\R^n</math> containing the origin (Here, symmetric means that if x is in the subset then -x is also in the subset).<ref>Template:Springer</ref>
More generally, if <math>M</math> is a compact n-dimensional Riemannian manifold, and <math>f: M \rightarrow \mathbb{R}^n</math> is continuous, there exists a pair of points x and y in <math>M</math> such that <math>f(x) = f(y)</math> and x and y are joined by a geodesic of length <math>\delta</math>, for any prescribed <math>\delta > 0</math>.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
Consider the function A which maps a point to its antipodal point: <math>A(x) = -x.</math> Note that <math>A(A(x))=x.</math> The original theorem claims that there is a point x in which <math>f(A(x))=f(x).</math> In general, this is true also for every function A for which <math>A(A(x))=x.</math><ref>Template:Cite journal</ref> However, in general this is not true for other functions A.<ref>{{#invoke:citation/CS1|citation

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NotesEdit

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ReferencesEdit

External linksEdit

Template:YouTube
The Borsuk-Ulam Explorer. An interactive illustration of Borsuk-Ulam Theorem.