论文标题
同源不变的可确定内容ii：čech共同体和同型分类
The definable content of homological invariants II: Čech cohomology and homotopy classification
论文作者
论文摘要
这是一系列论文中的第二部分，该论文采用了描述性的理论技术来分析和丰富来自同源代数和代数拓扑的经典函数。在其中，我们证明了čeCh的共同体函数$ \ check {\ mathrm {h}}^n $在本地紧凑的可分开度量空间的类别上，每个因子中的每个因素中的每个因素都在（i）我们称为其定义的版本中，函数$ \ \ \ \ check check {\ mathrm {\ mathrm {h}}}}}^n _ = $ \ mathsf {gpc} $具有波兰封面的组（本工作的前任首次引入的类别），其次是（ii）（ii）$ \ Mathsf {gpc} $的健忘函数到组类别。 These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of $d$-spheres or $d$-tori for any $d\geq 1$, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical协同学函数是恒定的。然后，我们应用函数$ \ check {\ mathrm {h}}^n _ {\ mathrm {def}} $，以表明代数拓扑的开发中的开创性问题，即borsuk和eilenberg和eilenberg的1936年1936年分类的问题，直至$ solotopy，直至$ sopempopy $ sole $ sopement $ sopement $ sopement $ sopement $ s^33 33 33 33 33 33 333.33。 $ 2 $ -Sphere，本质上是高限度但不光滑的。 在这项工作的过程中，我们记录了许多经典结果的可定义版本，这些版本均与čech共同体的组合和同位素配方相关。总体而言，这项工作可能被认为是从局部紧凑型波兰人到多面体的地图空间上的描述性设定理论研究的基础，这种关系体现了整个数学整个数学的各种分类问题。
This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors $\check{\mathrm{H}}^n$ on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor $\check{\mathrm{H}}^n_{\mathrm{def}}$ taking values in the category $\mathsf{GPC}$ of groups with a Polish cover (a category first introduced in this work's predecessor), followed by (ii) a forgetful functor from $\mathsf{GPC}$ to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of $d$-spheres or $d$-tori for any $d\geq 1$, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors $\check{\mathrm{H}}^n_{\mathrm{def}}$ to show that a seminal problem in the development of algebraic topology, namely Borsuk and Eilenberg's 1936 problem of classifying, up to homotopy, the maps from a solenoid complement $S^3\backslashΣ$ to the $2$-sphere, is essentially hyperfinite but not smooth. In the course of this work, we record Borel definable versions of a number of classical results bearing on both the combinatorial and homotopical formulations of Čech cohomology; in aggregate, this work may be regarded as laying foundations for the descriptive set theoretic study of the homotopy relation on the space of maps from a locally compact Polish space to a polyhedron, a relation which embodies a substantial variety of classification problems arising throughout mathematics.
