Homotopy groups and quantitative Sperner-type lemma

TL;DR

This paper generalizes Sperner's lemma using homotopy groups and establishes lower bounds on the number of fully colored simplices in a triangulation based on topological invariants like the Brouwer degree and Hopf invariant.

Contribution

It introduces a new invariant $mma(x)$ for homotopy classes and proves a lower bound on fully colored simplices using a generalized Pontryagin's theorem.

Findings

For $m=n$, $mma(x)$ equals the Brouwer degree.

For $m=3, n=2$, $mma(x)$ has a lower bound related to the Hopf invariant.

The number of fully colored $n$-simplices is at least $mma([f])$.

Abstract

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 $\pi_m(S^n)$. For any $x$ in this homotopy group we define a non-negative integer $\mu(x)$. 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 $\mu(x)$, where $x$ is the Hopf invariant, and proved that $\mu(1)=\mu(2)=9$. The main result of this paper is the theorem that the number of fully colored $n$-simplexes in $T$ is not less than $\mu([f])$. To prove this theorem we use a generalization of Pontryagin's theorem for manifolds with respect to their boundaries.