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].

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