Borsuk-Ulam's Theorem

The Borsuk-Ulam theorem states that every continuous function of a - sphere in the -dimensional Euclidean space maps a pair of antipodal points to the same point. (Two points of a sphere are called antipodal if they lie in exactly opposite directions from the center point.) ${\displaystyle n}$ ${\displaystyle S^{n}}$ ${\displaystyle n}$

The case is often explained by the fact that at any point in time a pair of antipodal points exist on the earth's surface with the same temperatures and the same air pressure. This assumes that temperature and air pressure are continuous functions. ${\displaystyle n=2}$

Borsuk-Ulam's theorem was conjectured by Stanisław Ulam and proved by Karol Borsuk in 1933 . It is possible to derive Brouwer's fixed point theorem in an elementary way from the Borsuk-Ulam theorem . There are various generalizations of the sentence, so that one speaks of sentences of the Borsuk-Ulam type.

statement

There are several equivalent formulations of the sentence:

• Let be a continuous antipodal map, then is . Antipodal means that applies to everyone . ${\displaystyle f\colon S^{n}\to S^{m}}$ ${\displaystyle n\leq m}$ ${\displaystyle f(-x)=-f(x)}$ ${\displaystyle x\in S^{n}}$
• Let be a continuous antipodal map. Then there is one with ${\displaystyle f\colon S^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle x\in S^{n}}$ ${\displaystyle f(x)=0}$
• Be a continuous map. Then there is a fixed point with . This is the formulation in the introduction. ${\displaystyle f\colon S^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle x\in S^{n}}$ ${\displaystyle f(x)=f(-x)}$
• If the n-sphere is covered by (n + 1) open or closed subsets of the n-sphere, at least one of the two contains an antipodal pair of points. ${\displaystyle S_{i}}$ ${\displaystyle S_{i}}$

Borsuk antipode theorem

A stronger statement is Borsuk's theorem, also known as Borsuk's antipodal theorem. A function is called antipode-preserving if it is odd .

statement

Is a symmetrical, open and bounded subset of des , which contains the zero point, and is continuous and antipode-preserving, i.e. for all , as well as . Then Brouwer's degree of mapping is an odd number. ${\displaystyle \Omega }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f\colon {\overline {\Omega }}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle f(-x)=-f(x)}$ ${\displaystyle x\in {\overline {\Omega }}}$ ${\displaystyle 0\not \in f(\partial \Omega )}$ ${\displaystyle d(f,\Omega ,0)}$

Further generalizations

• Instead of demanding that it be antipodal, it is enough ${\displaystyle f\in C({\overline {\Omega }},\mathbb {R} ^{n})}$

${\displaystyle {\frac {f(x)}{\|f(x)\|}}\neq {\frac {f(-x)}{\|f(-x)\|}}}$

and to demand. Functions that fulfill this are homotopic to an antipode-preserving function, which is sufficient for the proof of Borsuk's theorem. In particular, there is no continuous continuation from on with . Because if Brouwer's degree of mapping is not equal to zero, then the equation has at least one solution . ${\displaystyle 0\not \in f(\partial \Omega )}$ ${\displaystyle f|_{\partial \Omega }}$ ${\displaystyle \Omega }$ ${\displaystyle 0\notin f(\Omega )}$ ${\displaystyle f(x)=0}$ ${\displaystyle x\in \Omega }$

• The statement can also be generalized to infinitely dimensional normalized spaces . It should be a symmetrical, open and bounded subset of the normalized space , wherein a compact image is, and ${\displaystyle \Omega }$ ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle 0\in \Omega }$ ${\displaystyle F=\operatorname {Id} -F_{0}}$ ${\displaystyle F_{0}\colon {\overline {\Omega }}\to X}$ ${\displaystyle 0\not \in F(\partial \Omega )}$

${\displaystyle {\frac {F(x)}{\|F(x)\|}}\neq {\frac {F(-x)}{\|F(-x)\|}}.}$

Then the Leray shudder degree is an odd number.

application

In elementary geometry , the statement of Borsuk-Ulam can prove the following interesting fact (also known as Stone-Tukey's theorem or Ham sandwich theorem ):

"Given any two polygons in the plane. Then a straight line exists such that it halves the area of ​​both polygons at the same time . (ie not only in total, but even both separately) "

proof

Let be and denote the given polygons with. Consider this in the shifted - -plane that we consider in standard Euclidean space. Let then be the position vector of a point on the unit sphere and denote with the normal plane to through the zero point. For the intersection of with defines a straight line . This is precisely illustrations can be explained by virtue of the continuous mapping: . Apparently these images have the property . If the measure of a content denotes, another continuous mapping of can be explained with the definition . Borsuk-Ulam then provides for a point the existence of . According to the construction of , this point applies to both . The straight line we are looking for is thus from the assertion. ${\displaystyle i\in \{1,2\}}$ ${\displaystyle A_{i}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle D:=\mathbb {R} ^{2}\times \{1\}\subseteq \mathbb {R} ^{3}}$ ${\displaystyle u\in S^{2}}$ ${\displaystyle P_{u}}$ ${\displaystyle u}$ ${\displaystyle u\neq {\begin{pmatrix}0&0&\pm 1\end{pmatrix}}^{\top }}$ ${\displaystyle P_{u}}$ ${\displaystyle D}$ ${\displaystyle L_{u}}$ ${\displaystyle f_{i}\colon S^{2}\to \mathbb {R} ^{2}}$ ${\displaystyle u\mapsto {\text{Inhalt von }}A_{i}{\text{ in }}u{\text{-Richtung von }}L_{u}}$ ${\displaystyle f_{i}(u)+f_{i}(-u)={\text{Inhalt}}(A_{i})}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle F(u):=(\|f_{1}(u)\|,\|f_{2}(u)\|)^{\top }}$ ${\displaystyle S^{2}\to \mathbb {R} ^{2}}$ ${\displaystyle F}$ ${\displaystyle u^{*}}$ ${\displaystyle F(u^{*})=F(-u^{*})}$ ${\displaystyle F}$ ${\displaystyle \|f_{i}(u^{*})\|=\|f_{i}(-u^{*})\|={\tfrac {1}{2}}\|A_{i}\|}$ ${\displaystyle i\in \{1,2\}}$ ${\displaystyle L_{u^{*}}}$

The sentence is also used in topological combinatorics . There the theorem is closely related to Tucker's lemma and is equivalent to it. Sometimes the sentence of Borsuk-Ulam is used there in a variant or generalization by Albrecht Dold .

literature

• Karol Borsuk: Three sentences about the -dimensional Euclidean sphere ${\displaystyle n}$ . Fundamenta Mathematicae 20 (1933), 177-190, online
• Klaus Deimling: Nonlinear Functional Analysis . 1st edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 .
• Wolfgang Gromes: A simple proof of Borsuk's theorem . Mathematische Zeitschrift 178 (1981), 399-400, online .
• Lasar Ljusternik and Lew Schnirelmann : Topological Methods in Variational Problems . Issledowatelskii Institut Matematiki i Mechaniki pri OMGU, Moscow 1930 (Russian).
  French translation by J. Kravtchenko: Méthodes topologiques dans lesproblemèmes variationnels. 1ère partie. Espaces à un nombre fini de dimensions . Hermann & Cie., Paris 1934.
• Jiří Matoušek: Using the Borsuk-Ulam theorem . Springer-Verlag, Berlin 2003, ISBN 3-540-00362-2 .

Web links

• Allen Hatcher: Algebraic Topology (Eng.)
• Derivation of Brouwer's Fixed Point Theorem using the Borsuk-Ulam theorem (PDF file; 125 kB)
• Video: Borsuk antipodal theorem . Institute for Scientific Film (IWF) 1982, made available by the Technical Information Library (TIB), doi : 10.3203 / IWF / C-1433 .

Individual evidence

1. ↑ Mark de Longueville, A course in topological combinatorics, Springer 2013, p. 12
2. ^ Dold, Simple proofs of the Borsuk-Ulam results, Contemporary Mathematics, Volume 19, 1983, pp. 65-69