论文标题
可定义的Eilenberg--Mac Lane通用系数定理
Definable Eilenberg--Mac Lane Universal Coefficient Theorems
论文作者
论文摘要
我们证明了Eilenberg--Mac Lane的通用系数定理的可定义版本，这些版本在其整体共同体学组方面表达了一个紧凑的Metrizable空间的（Steenrod）同源性组，以及（CHECH）共同同源物组的多面体组的（čech）共同体学组。确切地说，我们表明，考虑到一个可固定的$ x $，A（不一定是紧凑的）多面体$ y $，以及带有分区关闭属性的Abelian Polish Group $ g $，有自然的确切序列\ Begin {equination*} 0 \ rightArrow \ rightArrow \ rightArrow \ rightrm {ext} ext} \ ext} \ ext} \ weft（ext） h_ {n}（x; g）\ rightarrow \ mathrm {hom} \ left（h^{n}（x）（x），g \ right）\ rightArrow 0 \ end {equation*}和\ begin {equination*} 0 \ rightArrow \ rightArrow \ rightArrow \ rightarrow \ mathrm {ext} h^{n}（y; g）\ rightarrow \ mathrm {hom} \ left（h_ {n}（y），g \ right）\ rightarrow 0 \ end {equation {equation {equation {equation*}，它一定会拆分，其中$ h_ {n}（n}（x; g）$是$ n $ dimdimenable $ n $ dimdimenable $ $ x $ $ x $ $ x $ x $ x $ x $ x $ x $ h^{n}（y; g）$是$ n $二维的$ y $ $ y $的可确定的共同体，并带有$ g $的系数。 这两个结果均以一般代数通用系数定理的推论获得，该定理将可数的自由Abelian群体的Cochain复合物与波兰群体的$ G $双链链复合物的可确定同源性相关联。
We prove definable versions of the Universal Coefficient Theorems of Eilenberg--Mac Lane expressing the (Steenrod) homology groups of a compact metrizable space in terms of its integral cohomology groups, and the (Čech) cohomology groups of a polyhedron in terms of its integral homology groups. Precisely, we show that, given a compact metrizable space $X$, a (not necessarily compact) polyhedron $Y$, and an abelian Polish group $G$ with the division closure property, there are natural definable exact sequences \begin{equation*} 0\rightarrow \mathrm{Ext}\left( H^{n+1}(X),G\right) \rightarrow H_{n}(X;G)\rightarrow \mathrm{Hom}\left( H^{n}(X),G\right) \rightarrow 0 \end{equation*} and \begin{equation*} 0\rightarrow \mathrm{Ext}\left( H_{n-1}(Y),G\right) \rightarrow H^{n}(Y;G)\rightarrow \mathrm{Hom}\left( H_{n}(Y),G\right) \rightarrow 0 \end{equation*} which definably split, where $H_{n}(X;G)$ is the $n$-dimensional definable homology group of $X$ with coefficients in $G$ and $H^{n}(Y;G)$ is the $n$ -dimensional definable cohomology group of $Y$ with coefficients in $G$. Both of these results are obtained as corollaries of a general algebraic Universal Coefficient Theorem relating the cohomology of a cochain complex of countable free abelian groups to the definable homology of its $G$-dual chain complex of Polish groups.
