Lusternik–Schnirelmann theorem

In mathematics, the Lusternik–Schnirelmann theorem, aka Lusternik–Schnirelmann–Borsuk theorem or LSB theorem, says as follows.

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.

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 topologyCombinatoricsSet covering
Brouwer fixed-point theoremSperner's lemmaKnaster–Kuratowski–Mazurkiewicz lemma
Borsuk–Ulam theoremTucker's lemmaLusternik–Schnirelmann theorem

References
  1. Bollobás, Béla (2006), The art of mathematics: Coffee time in Memphis, New York: Cambridge University Press, pp. 118–119, doi:10.1017/CBO9780511816574, ISBN 978-0-521-69395-0, MR 2285090.
  2. Lusternik, Lazar; Schnirelmann, Lev (1930), Méthodes topologiques dans les problèmes variationnels, Moscow: Gosudarstvennoe Izdat.. Bollobás (2006) cites pp. 26–31 of this 68-page pamphlet for the theorem.
  3. 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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