Homotopy groups and quantitative Sperner-type lemma
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866910870524657664 |
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| author | Musin, Oleg R. |
| author_facet | Musin, Oleg R. |
| contents | We consider a generalization of Sperner's lemma for a triangulation $T$ of $(m+1)$-discs $D$ whose vertices are colored in $n+2$ colors. A proper coloring of $T$ on the boundary of $D$ determines a simplicial mapping $f:S^m \to S^n$ and the element $x=[f]$ in $π_m(S^n)$. For any $x$ in this homotopy group we define a non-negative integer $μ(x)$. For some cases this invariant can be found explicitly. Namely, if $m=n$ then this number is the Brouwer degree of the mapping $f$. For the case $m=3, n=2$ we found a lower bound for $μ(x)$, where $x$ is the Hopf invariant, and proved that $μ(1)=μ(2)=9$.
The main result of this paper is the theorem that the number of fully colored $n$-simplexes in $T$ is not less than $μ([f])$. To prove this theorem we use a generalization of Pontryagin's theorem for manifolds with respect to their boundaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_08715 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Homotopy groups and quantitative Sperner-type lemma Musin, Oleg R. Algebraic Topology Combinatorics Geometric Topology 55M20, 55M25, 55P05 We consider a generalization of Sperner's lemma for a triangulation $T$ of $(m+1)$-discs $D$ whose vertices are colored in $n+2$ colors. A proper coloring of $T$ on the boundary of $D$ determines a simplicial mapping $f:S^m \to S^n$ and the element $x=[f]$ in $π_m(S^n)$. For any $x$ in this homotopy group we define a non-negative integer $μ(x)$. For some cases this invariant can be found explicitly. Namely, if $m=n$ then this number is the Brouwer degree of the mapping $f$. For the case $m=3, n=2$ we found a lower bound for $μ(x)$, where $x$ is the Hopf invariant, and proved that $μ(1)=μ(2)=9$. The main result of this paper is the theorem that the number of fully colored $n$-simplexes in $T$ is not less than $μ([f])$. To prove this theorem we use a generalization of Pontryagin's theorem for manifolds with respect to their boundaries. |
| title | Homotopy groups and quantitative Sperner-type lemma |
| topic | Algebraic Topology Combinatorics Geometric Topology 55M20, 55M25, 55P05 |
| url | https://arxiv.org/abs/2007.08715 |