Between Whitehead groups and uniformization

TL;DR

The paper demonstrates, in ZFC, an equivalence between a combinatorial ladder-system uniformization principle and a structural Ext-vanishing property for a broad class of abelian groups: specifically, every $S$-ladder system has $\aleph_0$-uniformization if and only if every strongly $\aleph_1$-free abelian group of size $\aleph_1$ with non-freeness confined to $S$ is $\aleph_1$-coseparable (i.e. $\operatorname{Ext}(G, \bigoplus_{i=0}^{\infty} \mathbb Z)=0$, and hence Whitehead). This resolves Eklof–Mekler problems B3 and B4 by translating the group-theoretic questions into combinatorial uniformization statements and proving a robust combinatorial equivalence (A_S $\Leftrightarrow$ B_S). The approach combines a detailed algebraic framework for $\aleph_1$-free and Whitehead groups with a combinatorial and forcing-based analysis of $S$-ladder systems, culminating in a unified mechanism to derive Whitehead properties from uniformization hypotheses and vice versa. The results provide a pathway to extend the analysis to Shelah groups and deepen the bridge between abelian group theory and ladder-system combinatorics.

Abstract

For a given stationary set $S$ of countable ordinals we prove (in $\mathbf{ZFC}$) that the assertion "every $S$-ladder system has $\aleph_0$-uniformization" is equivalent to "every strongly $\aleph_1$-free abelian group of cardinality $\aleph_1$ with non-freeness invariant $\subseteq S$ is $\aleph_1$-coseparable, i.e. Ext$(G, \oplus_{i=0}^{\infty} \mathbb Z)=0$ (in particular Whitehead, i.e.\ Ext$(G, \mathbb Z)=0$)". This solves problems B3 and B4 from Eklof and Mekler's monograph.

Theorems & Definitions (115)

• Definition 2
• Definition 3
• Definition 4
• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Corollary 1.4
• proof : Proof.
• Lemma 1.6
• proof : Proof.