On maps with continuous path lifting

IF 0.5 3区 数学 Q3 MATHEMATICS
Jeremy Brazas, A. Mitra
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引用次数: 0

Abstract

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 $\pi_1$: 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 $H\leq \pi_1(X,x_0)$ with totally path-disconnected coset space $\pi_1(X,x_0)/H$. Here, "weak equivalence" refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.
在具有连续路径提升的地图上
我们研究了用唯一提升性质定义的覆盖投影的自然推广。一张地图 $p:E\to X$ 有“连续路径覆盖属性”,如果所有的路径 $X$ 相对于紧开拓扑的唯一且连续的升力(即基点)。我们证明了具有这一性质的映射与具有完全路径断开的纤维的纤维和同伦群上的自然商拓扑密切相关。特别地,具有连续路径覆盖性质的映射类恰好位于Hurewicz纤维和具有完全路径断开纤维的Serre纤维之间。我们从拓扑学的角度将覆盖投影的分类扩展为具有连续路径覆盖性质的映射的分类 $\pi_1$:对于任何连通路径的Hausdorff空间 $X$、地图 $E\to X$ 利用连续路径覆盖的性质,通过子群将其划分为弱等价 $H\leq \pi_1(X,x_0)$ 具有完全无路径的余集空间 $\pi_1(X,x_0)/H$. 这里的“弱等价”是指由形式反演的双射弱同伦等价所生成的等价关系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Fundamenta Mathematicae
Fundamenta Mathematicae 数学-数学
CiteScore
1.00
自引率
0.00%
发文量
44
审稿时长
6-12 weeks
期刊介绍: FUNDAMENTA MATHEMATICAE concentrates on papers devoted to Set Theory, Mathematical Logic and Foundations of Mathematics, Topology and its Interactions with Algebra, Dynamical Systems.
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