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In the Borsuk–Ulam theorem (K. Borsuk, 1933 [[#References|[a2]]]), topological and symmetry properties are used for coincidence assertions for mappings defined on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b1203901.png" />-dimensional unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b1203902.png" />. Obviously, the following three versions of this result are equivalent:
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In the Borsuk–Ulam theorem (K. Borsuk, 1933 [[#References|[a2]]]), topological and symmetry properties are used for coincidence assertions for mappings defined on the $n$-dimensional unit sphere $S^n\subset\mathbf R^{n+1}$. Obviously, the following three versions of this result are equivalent:
  
1) For every continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b1203903.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b1203904.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b1203905.png" />.
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1) For every continuous mapping $f\colon S^n\to\mathbf R^n$, there exists an $x\in S^n$ with $f(x)=f(-x)$.
  
2) For every odd continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b1203906.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b1203907.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b1203908.png" />.
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2) For every odd continuous mapping $f\colon S^n\to\mathbf R^n$, there exists an $x\in S^n$ with $f(x)=0$.
  
3) If there exists an odd continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b1203909.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039010.png" />. The Borsuk–Ulam theorem is equivalent, among others, to the fact that odd continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039011.png" /> are essential (cf. [[Antipodes|Antipodes]]), to the [[Lyusternik–Shnirel'man–Borsuk covering theorem|Lyusternik–Shnirel'man–Borsuk covering theorem]] and to the Krein–Krasnosel'skii–Mil'man theorem on the existence of vectors  "orthogonal"  to a given linear subspace [[#References|[a3]]].
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3) If there exists an odd continuous mapping $f\colon S^n\to S^m$, then $m\geq n$. The Borsuk–Ulam theorem is equivalent, among others, to the fact that odd continuous mappings $f\colon S^n\to S^n$ are essential (cf. [[Antipodes|Antipodes]]), to the [[Lyusternik–Shnirel'man–Borsuk covering theorem|Lyusternik–Shnirel'man–Borsuk covering theorem]] and to the Krein–Krasnosel'skii–Mil'man theorem on the existence of vectors  "orthogonal"  to a given linear subspace [[#References|[a3]]].
  
 
The Borsuk–Ulam theorem remains true:
 
The Borsuk–Ulam theorem remains true:
  
a) if one replaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039012.png" /> by the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039013.png" /> of a bounded neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039015.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039016.png" />;
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a) if one replaces $S^n$ by the boundary $\partial U$ of a bounded neighbourhood $U\subset\mathrm R^{n+1}$ of $0$ with $U=-U$;
  
b) for continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039018.png" /> is the unit sphere in a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039021.png" />, a linear subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039023.png" /> a compact mapping (for versions 1) and 2)).
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b) for continuous mappings $f\colon S\to Y$, where $S$ is the unit sphere in a [[Banach space|Banach space]] $X$, $Y\subset X$, $Y\neq X$, a linear subspace of $X$ and $\id-f$ a compact mapping (for versions 1) and 2)).
  
 
For more general symmetries, the following extension of version 3) holds:
 
For more general symmetries, the following extension of version 3) holds:
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039025.png" /> be finite-dimensional orthogonal representations of a compact [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039026.png" />, such that for some prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039027.png" />, some subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039028.png" /> acts freely on the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039029.png" />. If there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039030.png" />-mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039032.png" />.
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Let $V$ and $W$ be finite-dimensional orthogonal representations of a compact [[Lie group|Lie group]] $G$, such that for some prime number $p$, some subgroup $H\cong\mathbf Z/p$ acts freely on the unit sphere $SV$. If there exists a $G$-mapping $f\colon SV\to SW$, then $\dim V\leq\dim W$.
  
 
For related results under weaker conditions, cf. [[#References|[a1]]]; for applications, cf. [[#References|[a4]]].
 
For related results under weaker conditions, cf. [[#References|[a1]]]; for applications, cf. [[#References|[a4]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Bartsch,  "On the existence of Borsuk–Ulam theorems"  ''Topology'' , '''31'''  (1992)  pp. 533–543</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Borsuk,  "Drei Sätze über die <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120390/b12039033.png" />-dimensionale Sphäre"  ''Fund. Math.'' , '''20'''  (1933)  pp. 177–190</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.G. Krein,  M.A. Krasnosel'skii,  D.P. Mil'man,  "On the defect numbers of linear operators in a Banach space and some geometrical questions"  ''Sb. Trud. Inst. Mat. Akad. Nauk Ukrain. SSR'' , '''11'''  (1948)  pp. 97–112  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Steinlein,  "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'''  (1985)  pp. 166–235</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Bartsch,  "On the existence of Borsuk–Ulam theorems"  ''Topology'' , '''31'''  (1992)  pp. 533–543</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Borsuk,  "Drei Sätze über die $n$-dimensionale Sphäre"  ''Fund. Math.'' , '''20'''  (1933)  pp. 177–190</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.G. Krein,  M.A. Krasnosel'skii,  D.P. Mil'man,  "On the defect numbers of linear operators in a Banach space and some geometrical questions"  ''Sb. Trud. Inst. Mat. Akad. Nauk Ukrain. SSR'' , '''11'''  (1948)  pp. 97–112  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Steinlein,  "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'''  (1985)  pp. 166–235</TD></TR></table>

Revision as of 21:49, 31 December 2018

In the Borsuk–Ulam theorem (K. Borsuk, 1933 [a2]), topological and symmetry properties are used for coincidence assertions for mappings defined on the $n$-dimensional unit sphere $S^n\subset\mathbf R^{n+1}$. Obviously, the following three versions of this result are equivalent:

1) For every continuous mapping $f\colon S^n\to\mathbf R^n$, there exists an $x\in S^n$ with $f(x)=f(-x)$.

2) For every odd continuous mapping $f\colon S^n\to\mathbf R^n$, there exists an $x\in S^n$ with $f(x)=0$.

3) If there exists an odd continuous mapping $f\colon S^n\to S^m$, then $m\geq n$. The Borsuk–Ulam theorem is equivalent, among others, to the fact that odd continuous mappings $f\colon S^n\to S^n$ are essential (cf. Antipodes), to the Lyusternik–Shnirel'man–Borsuk covering theorem and to the Krein–Krasnosel'skii–Mil'man theorem on the existence of vectors "orthogonal" to a given linear subspace [a3].

The Borsuk–Ulam theorem remains true:

a) if one replaces $S^n$ by the boundary $\partial U$ of a bounded neighbourhood $U\subset\mathrm R^{n+1}$ of $0$ with $U=-U$;

b) for continuous mappings $f\colon S\to Y$, where $S$ is the unit sphere in a Banach space $X$, $Y\subset X$, $Y\neq X$, a linear subspace of $X$ and $\id-f$ a compact mapping (for versions 1) and 2)).

For more general symmetries, the following extension of version 3) holds:

Let $V$ and $W$ be finite-dimensional orthogonal representations of a compact Lie group $G$, such that for some prime number $p$, some subgroup $H\cong\mathbf Z/p$ acts freely on the unit sphere $SV$. If there exists a $G$-mapping $f\colon SV\to SW$, then $\dim V\leq\dim W$.

For related results under weaker conditions, cf. [a1]; for applications, cf. [a4].

References

[a1] T. Bartsch, "On the existence of Borsuk–Ulam theorems" Topology , 31 (1992) pp. 533–543
[a2] K. Borsuk, "Drei Sätze über die $n$-dimensionale Sphäre" Fund. Math. , 20 (1933) pp. 177–190
[a3] M.G. Krein, M.A. Krasnosel'skii, D.P. Mil'man, "On the defect numbers of linear operators in a Banach space and some geometrical questions" Sb. Trud. Inst. Mat. Akad. Nauk Ukrain. SSR , 11 (1948) pp. 97–112 (In Russian)
[a4] H. Steinlein, "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 (1985) pp. 166–235
How to Cite This Entry:
Borsuk-Ulam theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borsuk-Ulam_theorem&oldid=43630
This article was adapted from an original article by H. Steinlein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article