Homotopy groups and quantitative Sperner-type lemma
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.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2020
- DOI:
- 10.48550/arXiv.2007.08715
- arXiv:
- arXiv:2007.08715
- Bibcode:
- 2020arXiv200708715M
- Keywords:
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- Mathematics - Algebraic Topology;
- Mathematics - Combinatorics;
- Mathematics - Geometric Topology
- E-Print:
- 10 pages, 1 figure