论文标题
同型公制空间组的天然假学计量学
A natural pseudometric on homotopy groups of metric spaces
论文作者
论文摘要
对于路径连接的度量空间$（x，d）$，$ n $ th同型组$π_n（x）$继承了带有统一度量的$ n $ th迭代环空间的天然伪测量方法。该伪计给出$π_n（x）$是拓扑组的结构，当$ x $紧凑时，诱导的伪测量学拓扑独立于公制$ d $。在本文中，我们研究了该伪计的特性及其与$π_n（x）$的先前研究结构的关系。我们的主要结果是，如果$ x $紧凑，$ x $且$ lc^{n-1} $或$ x $是有限的PolyHedra和retraction Bonding图的倒数倒数，那么假学拓扑与$π_n（x）$上的形状拓扑相符。
For a path-connected metric space $(X,d)$, the $n$-th homotopy group $π_n(X)$ inherits a natural pseudometric from the $n$-th iterated loop space with the uniform metric. This pseudometric gives $π_n(X)$ the structure of a topological group and when $X$ is compact, the induced pseudometric topology is independent of the metric $d$. In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on $π_n(X)$. Our main result is that the pseudometric topology agrees with the shape topology on $π_n(X)$ if $X$ is compact and $LC^{n-1}$ or if $X$ is an inverse limit of finite polyhedra with retraction bonding maps.
