Between Whitehead groups and uniformization

TL;DR

This paper establishes an equivalence in ZFC between a property of S-ladder systems and a class of abelian groups with certain freeness and extension properties, solving open problems in the theory of Whitehead groups.

Contribution

It proves the equivalence between uniformization of S-ladder systems and properties of strongly -free abelian groups, resolving key open problems in the field.

Findings

Equivalence between S-ladder system uniformization and Whitehead group properties

Resolution of problems B3 and B4 from Eklof and Mekler's monograph

Characterization of -coseparable abelian groups in terms of ladder systems

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.