Whitehead群的一致化与多样性
Uniformization and the diversity of Whitehead groups
摘要：
第三作者在[Sh：98]中研究了Whitehead群与单值化性质之间的联系。特别是，它基本上表明，有一个非自由的Whitehead（分别为aleph_1-coseparable）的基数aleph_1的群，当且仅当有一个阶梯系统上的一个固定子集的omega_1满足2-uniformization（分别为omega-uniformization）。这些技术也允许证明各种独立性和一致性的结果约Whitehead集团，例如，它是一致的，有一个非自由的Whitehead集团的基数aleph_1，但没有非自由aleph_1-coseparable集团。然而，一些自然问题仍然悬而未决，其中有以下两个：（i）基数aleph_1的W-群的类恰好是基数aleph_1的强aleph_1-自由群的类是否相容？(ii)如果基数为aleph_1的强aleph_1-自由群都是W-群，那么它们也都是aleph_1-余可分的吗？在本文中，我们使用的技术的一致化回答的第一个问题的否定和部分肯定的回答第二个问题。
英文摘要：
The connections between Whitehead groups and uniformization properties were investigated by the third author in [Sh:98]. In particular it was essentially shown there that there is a non-free Whitehead (respectively, aleph_1-coseparable) group of cardinality aleph_1 if and only if there is a ladder system on a stationary subset of omega_1 which satisfies 2-uniformization (respectively, omega-uniformization). These techniques allowed also the proof of various independence and consistency results about Whitehead groups, for example that it is consistent that there is a non-free Whitehead group of cardinality aleph_1 but no non-free aleph_1-coseparable group. However, some natural questions remained open, among them the following two: (i) Is it consistent that the class of W-groups of cardinality aleph_1 is exactly the class of strongly aleph_1-free groups of cardinality aleph_1 ? (ii) If every strongly aleph_1-free group of cardinality aleph_1 is a W-group, are they also all aleph_1-coseparable? In this paper we use the techniques of uniformization to answer the first question in the negative and give a partial affirmative answer to the second question.
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