The definable content of homological invariants I: Ext and lim1

This is the first installment in a series of papers illustrating how classical invariants of homological algebra and algebraic topology may be enriched with additional descriptive set theoretic information. To effect this enrichment, we show that many of these invariants may be naturally regarded as functors to the category, introduced herein, of \emph{groups with a Polish cover}. The resulting \emph{definable invariants} provide far stronger means of classification. In the present work we focus on the first derived functors of $\mathrm{Hom}(-,-)$ and $\mathrm{lim}(-)$. The resulting \emph{definable $\mathrm{Ext}(B,F)$} for pairs of countable abelian groups $B,F$ and \emph{definable $\mathrm{lim}^{1}(\boldsymbol{A})$} for towers $\boldsymbol{A}$ of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable $\textrm{Ext}( -,\mathbb{Z}% )$ is a fully faithful contravariant functor from the category of finite rank torsion-free abelian groups $\Lambda$ with no free summands; this contrasts with the fact that there are uncountably many non-isomorphic such groups $\Lambda$ with isomorphic classical invariants $\textrm{Ext}( \Lambda ,\mathbb{Z}) $. To facilitate our analysis, we introduce a general \emph{Ulam stability framework} for groups with a Polish cover; within this framework we prove several rigidity results for non-Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of $p$-adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem $\mathcal{R}(\mathrm{Aut}( \Lambda) \curvearrowright \mathrm{Ext}( \Lambda ,\mathbb{Z% }))$ of classifying all group extensions of $\Lambda$ by $\mathbb{Z}$ up to \emph{base-free isomorphism}, when $\Lambda =\mathbb{Z}[1/p]^{d}$ for prime numbers $p$ and $ d\geq 1$.