It is consistent with ZFC that B_1-groups are not B_2-groups

Abstract

A torsion-free abelian group B of arbitrary rank is called a B_1-group if Bext^1(B,T)=0 for every torsion abelian group T, where Bext^1 denotes the group of equivalence classes of all balanced exact extensions of T by B. It is a long-standing problem whether or not the class of B_1-groups coincides with the class of B_2-groups. A torsion-free abelian group B is called a B_2-group if there exists a continuous well-ordered ascending chain of pure subgroups, 0=B_0 subset B_1 subset ... subset B_alpha subset ... subset B_lambda =B=bigcup_{alpha in lambda} B_alpha such that B_{alpha+1}=B_alpha+G_alpha for every alpha in lambda for some finite rank Butler group G_alpha. Both, B_1-groups and B_2-groups are natural generalizations of finite rank Butler groups to the infinite rank case and it is known that every B_2-group is a B_1-group. Moreover, assuming V=L it was proven that the two classes coincide. Here we demonstrate that it is undecidable in ZFC whether or not all B_1-groups are B_2-groups. Using Cohen forcing we prove that there is a model of ZFC in which there exists a B_1-group that is not a B_2-group.