Borsuk–Ulam theorem
In mathematics, the Borsuk–Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.
According to (Matoušek 2003, p. 25), the first historical mention of the statement of this theorem appears in (Lyusternik 1930). The first proof was given by (Borsuk 1933), where the formulation of the problem was attributed to Ulam. Since then, many alternate proofs have been found out by various authors as collected in (Steinlein 1985).
The case n = 2 is often illustrated by saying that at any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures. This assumes that temperature and barometric pressure vary continuously.
A stronger statement related to Borsuk–Ulam theorem is that every antipode-preserving map f from Sn to itself has odd degree.
Corollaries
- No subset of Rn is homeomorphic to Sn.
- The Lusternik–Schnirelmann theorem: If the sphere Sn is covered by n + 1 open sets, then one of these sets contains a pair (x, −x) of antipodal points. (this is equivalent to the Borsuk–Ulam theorem)
- The Ham sandwich theorem: For any compact sets
in Rn we can always find a hyperplane dividing each of them into two subsets of equal measure. - The Brouwer fixed-point theorem (Matoušek 2003, p. 25; Su 1997).
See also
- Sperner's lemma
- Tucker's lemma
- Topological combinatorics
- Necklace splitting problem
- Kakutani's theorem (geometry)
- Isovariant
References
- Borsuk, K. (1933). "Drei Sätze über die n-dimensionale euklidische Sphäre". Fund. Math. 20: 177–190.
- Lyusternik, L.; Shnirel'man, S. (1930). "Topological Methods in Variational Problems". Issledowatelskii Institut Matematiki i Mechaniki pri O. M. G. U. (Moscow).
- Matoušek, Jiří (2003). Using the Borsuk–Ulam theorem. Berlin: Springer Verlag. doi:10.1007/978-3-540-76649-0. ISBN 3-540-00362-2.
- Steinlein, H. (1985). "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: 166–235.
- Su, Francis Edward (Nov. 1997). "Borsuk-Ulam Implies Brouwer: A Direct Construction". The American Mathematical Monthly 104 (9): 855–859. http://www.math.hmc.edu/~su/papers.dir/borsuk.pdf.