论文标题
起重空间的一般理论
General theory of lifting spaces
论文作者
论文摘要
在他的代数拓扑结构的古典教科书中，埃德温·史威尔（Edwin Spanier）在更一般的提升空间框架内（即Hurewicz纤维具有独特的路径属性），开发了覆盖空间的理论。除其他外，Spanier证明，对于每个空间$ x $，都存在一个通用的提升空间，但是，除非基本空间$ x $是半平均连接的，否则不必简单地连接。关于环球空间的基本群体到底是什么问题的问题。 起重空间的主要来源是在$ x $上覆盖空间的逆限制，或更一般地，在某些逆向空间的反向系统中汇聚到$ x $。每个度量空间$ x $都可以作为Polyhedra反向系统的限制获得，因此在系统上覆盖空间覆盖空间的逆限制在$ x $上均超过$ x $。它们与$ x $的几何形状（尤其是基本组）相似，因为Polyhedra上的覆盖空间与其基地的基本组有关。因此，举起空间似乎是自然替代品，用于覆盖空间范围内的概念，其本地特性不良。 在本文中，我们开发了提升空间的一般理论，并证明它们是由产品，逆限制和其他重要结构保存的。 We show that maps from $X$ to polyhedra give rise to coverings over $X$ and use that to prove that for a connected, locally path connected and paracompact $X$, the fundamental group of the above-mentioned Spanier's universal space is precisely the intersection of all Spanier groups associated to open covers of $X$, and that the later coincides with the shape kernel of $X$.此外，我们更详细地检查了在$ x $上提升空间，这是覆盖物的反向限制，而这些近似值是$ x $的近似值。
In his classical textbook on algebraic topology Edwin Spanier developed the theory of covering spaces within a more general framework of lifting spaces (i.e., Hurewicz fibrations with unique path-lifting property). Among other, Spanier proved that for every space $X$ there exists a universal lifting space, which however need not be simply connected, unless the base space $X$ is semi-locally simply connected. The question on what exactly is the fundamental group of the universal space was left unanswered. The main source of lifting spaces are inverse limits of covering spaces over $X$, or more generally, over some inverse system of spaces converging to $X$. Every metric space $X$ can be obtained as a limit of an inverse system of polyhedra, and so inverse limits of covering spaces over the system yield lifting spaces over $X$. They are related to the geometry (in particular the fundamental group) of $X$ in a similar way as the covering spaces over polyhedra are related to the fundamental group of their base. Thus lifting spaces appear as a natural replacement for the concept of covering spaces over base spaces with bad local properties. In this paper we develop a general theory of lifting spaces and prove that they are preserved by products, inverse limits and other important constructions. We show that maps from $X$ to polyhedra give rise to coverings over $X$ and use that to prove that for a connected, locally path connected and paracompact $X$, the fundamental group of the above-mentioned Spanier's universal space is precisely the intersection of all Spanier groups associated to open covers of $X$, and that the later coincides with the shape kernel of $X$. Furthermore, we examine in more detail lifting spaces over $X$ that arise as inverse limits of coverings over some approximations of $X$.
