论文标题
同型组和定量的斑点型引理
Homotopy groups and quantitative Sperner-type lemma
论文作者
论文摘要
我们考虑对sperner的引理的概括为三角剖分$ t $ $（m+1）$ - 碟片$ d $，其顶点的颜色为$ n+2 $颜色。 $ d $的边界上的$ t $的适当着色确定简单映射$ f：s^m \ to s^n $，元素$ x = [f] $ in $π_m（s^n）$。对于此同质组中的任何$ x $，我们定义一个非负整数$μ（x）$。在某些情况下，可以明确找到此不变的。也就是说，如果$ m = n $，则此数字是映射$ f $的brouwer学位。对于$ m = 3，n = 2 $，我们找到了$μ（x）$的下限，其中$ x $是hopf不变的，并证明$μ（1）=μ（2）= 9 $。 本文的主要结果是定理，$ t $中的完全颜色的$ n $ simplexes数量不少于$μ（[f]）$。为了证明这一定理，我们将pontryagin定理的概括用于流形方面。
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.
