论文标题
在与连续路径提升的地图上
On maps with continuous path lifting
论文作者
论文摘要
我们研究根据独特的举重特性定义的覆盖预测的自然概括。地图$ p：e \ to x $具有“连续路径覆盖属性”，如果相对于紧凑型拓扑的$ x $ lift中的所有路径唯一，连续（rel。basepoint）。我们表明，与该属性的地图与具有完全途径的纤维以及同型组上的自然商拓扑密切相关。特别是，具有连续路径覆盖特性的地图类别在Hurewicz纤维纤维和带有完全路径连接的纤维的螺旋纤维之间正确地位于。 We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological $π_1$: for any path-connected Hausdorff space $X$, maps $E\to X$ with the continuous path-covering property are classified up to weak equivalence by subgroups $H\leq π_1(X,x_0)$ with totally path-disconnected coset space $π_1（x，x_0）/h $。在这里，“弱等价”是指正式反转弱弱同质等效产生的等价关系。
We study a natural generalization of covering projections defined in terms of unique lifting properties. A map $p:E\to X$ has the "continuous path-covering property" if all paths in $X$ lift uniquely and continuously (rel. basepoint) with respect to the compact-open topology. We show that maps with this property are closely related to fibrations with totally path-disconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological $π_1$: for any path-connected Hausdorff space $X$, maps $E\to X$ with the continuous path-covering property are classified up to weak equivalence by subgroups $H\leq π_1(X,x_0)$ with totally path-disconnected coset space $π_1(X,x_0)/H$. Here, "weak equivalence" refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.
