Authors

Oleg R. Musin

Document Type

Article

Publication Date

7-2020

Abstract

We consider a generalization of Sperner's lemma for triangulations of m-discs whose vertices are colored in at most m colors. A coloring on the boundary (m-1)-sphere defines an element in the corresponding homotopy group of the sphere. Depending on this invariant, a lower bound is obtained for the number of fully colored simplexes. In particular, if the Hopf invariant is nonzero on the boundary of 4-disk, then there are at least 9 fully colored tetrahedra and if the Hopf invariant is d, then the lower bound is 3d + 3.

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Mathematics Commons

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