A non-reflexive Whitehead group (Q5928855)

This paper is motivated by a theorem and a question due to M. Huber. He proved in ZFC that every \(\aleph_1\)-coseparable group is reflexive. He asked whether it is provable in ZFC that every Whitehead group is reflexive. This is true in any model where every Whitehead group is free. It is also true for Whitehead groups of cardinality \(\aleph_1\) in a model of MA + \(\neg\)CH. It was left as an open question whether every Whitehead group is reflexive.NEWLINENEWLINENEWLINEIn this paper the authors give a strong negative answer: Theorem 0.1. It is consistent with ZFC that there is a strongly non-reflexive strongly \(\aleph_1\)-free Whitehead group of cardinality \(\aleph_1\).NEWLINENEWLINENEWLINEMoreover it is proved Theorem 0.2. It is consistent with ZFC that there is a non-free strongly \(\aleph_1\)-free group of cardinality \(\aleph_1\) such that \(\text{Ext}(A,\mathbb{Z})\) is torsion and \(\text{Hom}(A,\mathbb{Z})=0\).NEWLINENEWLINENEWLINEFinally new examples of possible co-Moore spaces are provided, answering a recent question of Golasiński, Gonçalves.NEWLINENEWLINENEWLINEReviewer's remark: The models for both theorems result from a finite support iteration of c.c.c. posets and are models in ZFC + \(\neg\)CH.