2020/08/20 by Jeffrey Bergfalk, Martino Lupini, Bergfalk, Jeffrey +3
Mathematics · #18G10 #18G60 #37A20 #55N07 #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Logic (math.LO) #Primary 54H05 #Secondary 55N05
paper · pdf · doi:10.48550/arxiv.2008.08782
openalex publication_date 2020/08/20 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
This is the first installment in a series of papers in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information. To effect this enrichment, we show that many of these invariants can be naturally regarded as functors to the category, introduced herein, of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification. In the present work we focus on the first derived functors of Hom(-,-) and lim(-). The resulting definable Ext(B,F) for pairs of countable abelian groups B,F and definable lim1(\boldsymbolA) for towers \boldsymbolA of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable \textrmExt(-,ℤ) is a fully faithful contravariant functor from the category of finite rank torsion-free abelian groups Λ with no free summands; this contrasts with the fact that there are uncountably many non-isomorphic such groups Λ with isomorphic classical invariants \textrmExt(Λ,ℤ) . To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover and 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 the 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 R(Aut(Λ)\curvearrowrightExt(Λ,ℤ)) of classifying all group extensions of Λ by ℤ up to base-free isomorphism, when Λ=ℤ[1/p]d for prime numbers p and d≥ 1.