# Borsuk-Ulam theorem

(Redirected from Borsuk–Ulam theorem)

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].

How to Cite This Entry:
Borsuk–Ulam theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borsuk%E2%80%93Ulam_theorem&oldid=22172