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 -dimensional unit sphere . Obviously, the following three versions of this result are equivalent:

1) For every continuous mapping , there exists an with .

2) For every odd continuous mapping , there exists an with .

3) If there exists an odd continuous mapping , then . The Borsuk–Ulam theorem is equivalent, among others, to the fact that odd continuous mappings 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 by the boundary of a bounded neighbourhood of with ;

b) for continuous mappings , where is the unit sphere in a Banach space , , , a linear subspace of and a compact mapping (for versions 1) and 2)).

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

Let and be finite-dimensional orthogonal representations of a compact Lie group , such that for some prime number , some subgroup acts freely on the unit sphere . If there exists a -mapping , then .

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 -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