[Paper Review] The definable content of homological invariants I: $\mathrm{Ext}$ & $\mathrm{lim}^1$

Jeffrey Bergfalk, Martino Lupini+1 more|arXiv (Cornell University)|Aug 20, 2020

Advanced Topology and Set Theory30 references4 citations

TL;DR

This paper introduces a definable refinement of classical homological invariants—specifically Ext(B, F) and lim¹(A)—by viewing them as functors into the category of groups with a Polish cover, where morphisms are definable homomorphisms. It establishes that the definable Ext(−, ℤ) is fully faithful on finite rank torsion-free abelian groups without free summands, resolving a cardinality gap in classical invariants and answering a question of Kanovei and Reeken on p-adic quotients.

ABSTRACT

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 $\mathrm{Hom}(-,-)$ and $\mathrm{lim}(-)$. The resulting definable $\mathrm{Ext}(B,F)$ for pairs of countable abelian groups $B,F$ and 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 $ extrm{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 $ extrm{Ext}(\Lambda,\mathbb{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 $\mathcal{R}(\mathrm{Aut}(\Lambda)\curvearrowright\mathrm{Ext}(\Lambda,\mathbb{Z}))$ of classifying all group extensions of $\Lambda$ by $\mathbb{Z}$ up to base-free isomorphism, when $\Lambda =\mathbb{Z}[1/p]^{d}$ for prime numbers $p$ and $ d\geq 1$.

Motivation & Objective

• To enrich classical homological invariants like Ext and lim¹ with descriptive set-theoretic structure by embedding them in the category of groups with a Polish cover.
• To address the limitation of classical invariants, which fail to distinguish uncountably many non-isomorphic finite rank torsion-free abelian groups with isomorphic Ext(Λ, ℤ).
• To establish rigidity results for non-Archimedean abelian groups with a Polish cover using a new Ulam stability framework.
• To analyze the Borel reduction complexity of classifying group extensions via automorphism actions on Ext(Λ, ℤ), particularly for Λ = ℤ[1/p]^d.
• To resolve a question of Kanovei and Reeken concerning quotients of p-adic groups by showing that certain definable homomorphisms are trivial under stability conditions.

Proposed method

• Introduce the category of groups with a Polish cover, where objects are quotients G/N with G Polish and N Polishable, and morphisms are definable homomorphisms lifting to Borel maps.
• Develop a general Ulam stability framework for non-Archimedean abelian groups with a Polish cover, enabling rigidity results under topological and algebraic constraints.
• Apply cocycle superrigidity theorems for profinite actions of property (T) groups to analyze the complexity of classification problems in Ext(Λ, ℤ).
• Use the fact that definable homomorphisms lift to Borel maps, allowing a finer analysis of isomorphism types than classical group structure alone.
• Leverage the structure of profinite groups and their actions on Q_p^d to reduce classification problems to known Borel reducibility hierarchies.
• Prove that certain actions on orbit equivalence relations are ergodic and homotopic to constant maps, implying that nontrivial definable homomorphisms cannot exist under strong rigidity conditions.

Experimental results

Research questions

• RQ1 Can classical homological invariants like Ext(B, F) and lim¹(A) be refined using descriptive set theory to distinguish more isomorphism types?
• RQ2 Is the definable Ext(−, ℤ) functor fully faithful on the category of finite rank torsion-free abelian groups without free summands?
• RQ3 What is the Borel reduction complexity of classifying extensions of ℤ by Λ = ℤ[1/p]^d up to base-free isomorphism?
• RQ4 Do definable homomorphisms from orbit equivalence relations of certain profinite group actions to others necessarily factor through trivial maps under rigidity assumptions?
• RQ5 Can the Ulam stability framework for Polish-covered groups resolve open questions about quotients of p-adic groups, such as those posed by Kanovei and Reeken?

Key findings

• The definable Ext(−, ℤ) is a fully faithful contravariant functor on the category of finite rank torsion-free abelian groups with no free summands, resolving the issue that uncountably many such groups have isomorphic classical Ext(Λ, ℤ).
• A special case of the main rigidity result answers a question of Kanovei and Reeken: any definable homomorphism from a p-adic group quotient to a q-adic group quotient (for distinct primes p, q) is trivial if the source is a pro-p group and the target is a pro-q group.
• For Λ = ℤ[1/p]^d with d ≥ 1 and prime p, the action Aut(Λ) ↷ Ext(Λ, ℤ) has a definable classification problem whose complexity is bounded by the E_0 equivalence relation, and the orbit equivalence relation is E_0-ergodic.
• The orbit equivalence relation R(Γ ⋉ ℤ[1/q]^m ↷ ℚ_q^m) is F-ergodic for Γ ≤ SL_m(ℤ) with property (T) and m ≥ 3, showing high complexity in the Borel reducibility hierarchy.
• Using cocycle superrigidity, it is shown that any a.e. homomorphism from R(Γ ⋉ p^nℤ^m ↷ p^nℤ_p^m) to R(Δ ⋉ ℚ^d ↷ ℚ_q^d) is homotopic to a constant map when Γ has property (T), Δ ≤ GL_d(ℚ), and p ≠ q.
• The definable homomorphism category allows a finer classification of group extensions than the classical group-theoretic approach, particularly in the context of profinite group actions and Ulam stability.