Definable Eilenberg--Mac Lane Universal Coefficient Theorems

Abstract

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 (\v{C}ech) 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.