2020/07/17 by Oleg R. Musin, Musin, Oleg R. · 3 citations
Mathematics · #55M20 #55M25 #55P05 #Advanced Topics in Algebra #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Functional Equations Stability Results #Geometric Topology (math.GT)
paper · pdf · doi:10.48550/arxiv.2007.08715
openalex publication_date 2020/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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:Sm → Sn and the element x=[f] in πm(Sn). 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.