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Homotopy groups and quantitative Sperner-type lemma

Oleg R. Musin

TL;DR

The paper develops a quantitative Sperner-type framework by associating boundary colorings of triangulated disks with maps $f:S^m\to S^n$ and a homotopy-invariant class $x=[f]\in\pi_m(S^n)$, along with an invariant $\\mu(x)$. It proves a lower bound on the number of fully colored $n$-simplices in terms of $\\mu([f])$ by leveraging a Pontryagin-type correspondence between framed cobordisms and homotopy classes, unifying degree and Hopf-invariant cases. Concrete results include $\\mu(n,d)\ge d$ with sharp realizations in several dimensions, and $\\mu(1)=\\mu(2)=9$ for Hopf-invariant cases, plus a detailed study of the Hopf invariant through tetrahedral chains. The work also develops a relative-framed cobordism viewpoint and outlines several open problems on minimal triangulations and precise values of $\\mu$ for higher homotopy groups, highlighting connections between combinatorial topology and classical homotopy theory.

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 $π_m(S^n)$. 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.

Homotopy groups and quantitative Sperner-type lemma

TL;DR

The paper develops a quantitative Sperner-type framework by associating boundary colorings of triangulated disks with maps and a homotopy-invariant class , along with an invariant . It proves a lower bound on the number of fully colored -simplices in terms of by leveraging a Pontryagin-type correspondence between framed cobordisms and homotopy classes, unifying degree and Hopf-invariant cases. Concrete results include with sharp realizations in several dimensions, and for Hopf-invariant cases, plus a detailed study of the Hopf invariant through tetrahedral chains. The work also develops a relative-framed cobordism viewpoint and outlines several open problems on minimal triangulations and precise values of for higher homotopy groups, highlighting connections between combinatorial topology and classical homotopy theory.

Abstract

We consider a generalization of Sperner's lemma for a triangulation of -discs whose vertices are colored in colors. A proper coloring of on the boundary of determines a simplicial mapping and the element in . For any in this homotopy group we define a non-negative integer . For some cases this invariant can be found explicitly. Namely, if then this number is the Brouwer degree of the mapping . For the case we found a lower bound for , where is the Hopf invariant, and proved that . The main result of this paper is the theorem that the number of fully colored -simplexes in is not less than . To prove this theorem we use a generalization of Pontryagin's theorem for manifolds with respect to their boundaries.

Paper Structure

This paper contains 14 sections, 10 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sperner's lemma
  3. Homotopy invariants and Sperner's lemma
  4. $\mu$--invariant
  5. The degree of a map and $\mu$--invariant.
  6. Hopf invariant and tetrahedral chains
  7. Framed cobordisms and homotopy group of spheres
  8. Proof of the main theorem
  9. Concluding remarks and open problems
  10. Minimal simplicial maps of degree $d$.
  11. The lower bound on $\mu(d)$.
  12. Minimal simplicial map of Hopf invariant two.
  13. Find $\mu(n+1,n,x)$.
  14. Find $\mu(n+2,n,x)$.

Key Result

Theorem 1.1

Let $T$ be a triangulation of $D^4$ with a labeling $L:\mathop{\rm Vert}\nolimits(T)\to\{A,B,C,D\}$ such that $T$ has no fully labelled 3--simplices on its boundary $\partial T\cong S^3$. Let $\mathop{\rm \partial f_L}\nolimits$ on $\partial T$ be of Hopf invariant $d\ne0$. Then $T$ must contain at

Figures (1)

  • Figure 1: An illustration of Theorem A with $d=3$

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • ...and 6 more