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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Tucker-Ky Fan colorings


Author: K. S. Sarkaria
Journal: Proc. Amer. Math. Soc. 110 (1990), 1075-1081
MSC: Primary 57N10; Secondary 05C15, 57M05
MathSciNet review: 1004424
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Abstract: The existence of a continuous $ {{\mathbf{Z}}_2}$-map from a free $ m$-dimensional $ {{\mathbf{Z}}_2}$-simplicial complex $ E$ to the $ (m - 1)$-dimensional antipodal sphere $ {S^{m - 1}}$ is characterized by means of an enumerative combinatorial criterion involving a coloring of the vertices of $ E$. The Borsuk-Ulam theorem, 1933, and the combinatorial lemmas of Tucker, 1945, and Ky Fan, 1952, are easy consequences of this result for the case $ \vert E\vert = {S^m}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1004424-7
PII: S 0002-9939(1990)1004424-7
Article copyright: © Copyright 1990 American Mathematical Society