论文标题
较高同质组的元素无法通过多面体近似值检测到
Elements of higher homotopy groups undetectable by polyhedral approximation
论文作者
论文摘要
当拓扑空间$ x $中存在非平凡的本地结构时，一种表征$ n $ th同型同型同构类型的一种常见方法是$π_n（x，x_0）$的图像，是考虑$π_n（x，x_0）$ of $ n $ n $ n $ n $ - th的同型$ homom \ homom \ checkon $ \ checkon（x_0） $ψ_{n}：π_n（x，x_0）\ to \checkπ_n（x，x_0）$。子组$ \ ker（ψ_n）$是这种策略的阻塞，因为它由$π_n（x，x_0）$的元素组成，而多面体近似值至$ x $。在本文中，我们使用Spanier组的较高维度类似物来表征$ \ ker（ψ_n）$。特别是，我们证明，如果$ x $是paracompact，hausdorff和$ lc^{n-1} $，则$ \ ker（ψ_n）$等于$ n $ th $ th $ x $ x $。我们还利用更高的班级组的角度来概括Kozlowski-Segal的定理，该定理提供了条件，以确保$ψ_{n} $是同构的。
When non-trivial local structures are present in a topological space $X$, a common approach to characterizing the isomorphism type of the $n$-th homotopy group $π_n(X,x_0)$ is to consider the image of $π_n(X,x_0)$ in the $n$-th Čech homotopy group $\checkπ_n(X,x_0)$ under the canonical homomorphism $Ψ_{n}:π_n(X,x_0)\to \checkπ_n(X,x_0)$. The subgroup $\ker(Ψ_n)$ is the obstruction to this tactic as it consists of precisely those elements of $π_n(X,x_0)$, which cannot be detected by polyhedral approximations to $X$. In this paper, we use higher dimensional analogues of Spanier groups to characterize $\ker(Ψ_n)$. In particular, we prove that if $X$ is paracompact, Hausdorff, and $LC^{n-1}$, then $\ker(Ψ_n)$ is equal to the $n$-th Spanier group of $X$. We also use the perspective of higher Spanier groups to generalize a theorem of Kozlowski-Segal, which gives conditions ensuring that $Ψ_{n}$ is an isomorphism.
