Difference between revisions of "Borsuk-Ulam theorem"
From Encyclopedia of Mathematics
(Importing text file) |
|||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| − | In the Borsuk–Ulam theorem (K. Borsuk, 1933 [[#References|[a2]]]), topological and symmetry properties are used for coincidence assertions for mappings defined on the | + | {{TEX|done}} |
| + | 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 | + | 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 | + | 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 | + | 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 | + | a) if one replaces $S^n$ by the boundary $\partial U$ of a bounded neighbourhood $U\subset\mathbf R^{n+1}$ of $0$ with $U=-U$; |
| − | b) for continuous mappings | + | 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 | + | 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 | + | <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> |
Latest revision as of 21:50, 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 $U\subset\mathbf R^{n+1}$ 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=16557
Borsuk-Ulam theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borsuk-Ulam_theorem&oldid=16557
This article was adapted from an original article by H. Steinlein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article