Document Type
Article
Department
Mathematics (HMC)
Publication Date
4-2013
Abstract
We show that Fan’s 1952 lemma on labelled triangulations of the n-sphere with n + 1 labels is equivalent to the Borsuk–Ulam theorem. Moreover, unlike other Borsuk–Ulam equivalents, we show that this lemma directly implies Sperner’s Lemma, so this proof may be regarded as a combinatorial version of the fact that the Borsuk–Ulam theorem implies the Brouwer fixed-point theorem, or that the Lusternik–Schnirelmann–Borsuk theorem implies the KKM lemma.
Rights Information
© 2013 Mathematical Association of America. All Rights Reserved.
DOI
10.4169/amer.math.monthly.120.04.346
Recommended Citation
Nyman, Kathryn and Francis Edward Su. "A Borsuk-Ulam equivalent that implies Sperner’s Lemma." American Mathematical Monthly, 120 (2013), 346–354.
Comments
This article is also available from the American Mathematical Monthly at: http://www.maa.org/press/periodicals/american-mathematical-monthly/american-mathematical-monthly-contents-april-2013