Difference between revisions of "Whitehead problem"

A problem attributed, to J.H.C. Whitehead, which asks for a characterization of Abelian groups $A$ that satisfy the homological condition ${ \mathop{\rm Ext} } ( A, \mathbf Z ) = 0$, where $\mathbf Z$ is the group of integers under addition (cf. also Homology). A group which satisfies this condition is called a Whitehead group. An equivalent characterization is: $A$ is a Whitehead group if and only if for every surjective homomorphism $\pi$ onto $A$, if the kernel of $\pi$ is isomorphic to $\mathbf Z$, then the kernel is a direct summand of the domain of $\pi$. A sufficient condition for $A$ to be a Whitehead group is that $A$ is free (see Free Abelian group). This condition has been proved to be necessary if $A$ is countable. Since a subgroup of a Whitehead group is again a Whitehead group, this means that every Whitehead group is $\aleph _ {1}$- free, that is, every countable subgroup is free. Before 1973 only partial results were obtained for uncountable groups: Whitehead groups were proved to be separable and slender, and under the assumption of the continuum hypothesis ( ${ \mathop{\rm CH} }$), they were proved to be strongly $\aleph _ {1}$- free, that is, every countable subset is contained in a countable free subgroup which is a direct summand of countable subgroups containing it (see [a3] for results prior to 1973).

In 1973, S. Shelah [a5] proved that it is undecidable under the axioms of Zermelo–Frankel set theory (cf. also Set theory), ZFC, whether every Whitehead group of cardinality $\aleph _ {1}$ is free. Specifically, he proved that this is true assuming the Gödel constructibility axiom, $V = L$( see Gödel constructive set), but it is false assuming Martin's axiom (cf. Suslin hypothesis) and the negation of the continuum hypothesis ( ${ \mathop{\rm MA} } + \neg { \mathop{\rm CH} }$). Later he proved that $V = L$ implies that every Whitehead group, of arbitrary cardinality, is free [a6]. He also proved that the problem is undecidable even if ${ \mathop{\rm CH} }$ is assumed [a7] and that it is consistent with $\neg { \mathop{\rm CH} }$ that there are Whitehead groups of cardinality $\aleph _ {1}$ that are not strongly $\aleph _ {1}$- free [a8].