Lusternik–Schnirelmann theorem
This article is about the theorem on open covers of spheres. For the theorem on simple closed geodesics on spheres, see theorem of the three geodesics.
In mathematics, the Lusternik–Schnirelmann theorem, aka Lusternik–Schnirelmann–Borsuk theorem or LSB theorem, says as follows.
If the sphere Sn is covered by n + 1 open sets, then one of these sets contains a pair (x, −x) of antipodal points.
It is named after Lazar Lyusternik and Lev Schnirelmann, who published it in 1930.[1][2]
Equivalent results
There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in the top row can be deduced from the one below it in the same column.[3]
| Algebraic topology | Combinatorics | Set covering |
|---|---|---|
| Brouwer fixed-point theorem | Sperner's lemma | Knaster–Kuratowski–Mazurkiewicz lemma |
| Borsuk–Ulam theorem | Tucker's lemma | Lusternik–Schnirelmann theorem |
References
- ↑ Bollobás, Béla (2006), The art of mathematics: Coffee time in Memphis, Cambridge University Press, New York, pp. 118–119, doi:10.1017/CBO9780511816574, ISBN 978-0-521-69395-0, MR 2285090.
- ↑ Lusternik, L.; Schnirelmann, L. (1930), Méthodes topologiques dans les problèmes variationnels, Moscow: Gosudarstvennoe Izdat.. Bollobás (2006) cites pp. 26–31 of this 68-page pamphlet for the theorem.
- ↑ Nyman, Kathryn L.; Su, Francis Edward (2013), "A Borsuk–Ulam equivalent that directly implies Sperner's lemma", American Mathematical Monthly, 120 (4): 346–354, doi:10.4169/amer.math.monthly.120.04.346, MR 3035127
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