From: Allan Adler <ara@nestle.ai.mit.edu>
Subject: Re: structure of arbitrary abelian groups
Date: 03 Jan 2000 13:32:08 -0500
Newsgroups: sci.math
Summary: Whitehead conjecture on abelian groups

Allan Adler <ara@nestle.ai.mit.edu> writes:

> My recollection could be faulty on this, but I vaguely recall that
> one of the striking parts of Shelah's proof of the independence of the
> Whitehead conjecture involves exhibiting an explicit exact sequence
> 0 -> Z -> A -> G -> 0 of abelian groups, where Z is the additive group
> of integers, such that it is independent of ZFC whether the sequence
> splits or not.

Thanks to Matt Wiener for pointing out to me that I may have
misstated the situation.

I learned most of what I know about Shelah's proof from reading
Rotman's preprint decades ago, "Shelah's solution of Whitehead's
problem". I just took another look a it.

Here is what I know:
(1) Call an abelian group G aleph-1-free if every countable subgroup of G
    is a free abelian group. Aleph-1-free groups of cardinality aleph-1
    are partitioned into three types I,II,III. Those of type III are
    precisely the free ones.
(2) There exist groups of all three types. In particular, there is an
    explicit example of a group of type II. (See below)
(3) Call an abelian group G a W-group of every exact sequence
    0->Z->G->A->0 of abelian groups, where Z is the additive group
    of integers, splits.
(4) Assuming the axiom of constructibility, every W-group of cardinality
    aleph-1 is free. Assuming martin's axiom and the negation of the
    continuum hypothesis, every group of type II or III is a W-group.
(5) In particular, for the explicitly given group G of type II, G is
    a W-group if MA+not-CH and G is not a W-group if V=L.
(6) Thus, there is an explicit group G such that it is independent of ZFC
    whether G is a W-group. However, that is not as strong as saying that
    there is an exact sequence 0->Z->G->A->0 whose splitting is independent
    of ZFC. The proof that G is not a W-group if V=L constructs an exact
    sequence 0->Z->G->A->0 that doesn't split but that construction
    really uses V=L.

So I don't know whether the stronger assertion is true. For the explicitly
given aleph-1-free group of cardinality aleph-1 of type II, is there an
explicit exact sequence 0->Z->G->A->0 whose splitting is independent of ZFC?

For the record, here is the explicit construction of G, based on Rotman's
preprint.

Let W denote the first uncountable ordinal. Let U denote the additive group
of all functions from W to the additive group Z of integers. For each
countable limit ordinal d, choose an increasing sequence t(d) of
ordinals converging to d. For each such d and every positive integer n,
let x(d,n) be the function from W to Z supported on t(d) and whose
value on the m-th element of t(d) is m!/n! if m >= n and 0 otherwise.
Then G is the subgroup of U generated by all of the x(d,n) and all
functions with finite support.

Allan Adler
ara@zurich.ai.mit.edu

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