Worked solution: Shelah's proof that the Whitehead problem is independent of ZFC (1974)

Step 1 of 8: Free abelian groups: the simplest infinite building blocks

In plain words

Think of a free abelian group as an unlimited toolbox of independent directions you can walk in, combined using whole-number steps — like the familiar $\mathbb{Z}^n$ , but possibly with infinitely many independent directions instead of just finitely many. Every element is a unique, finite integer combination of a fixed "basis", and no unexpected relation between basis elements ever sneaks in.

This rigidity — a basis exists and every element decomposes over it uniquely — is an extremely strong property that most abelian groups do not have. J.H.C. Whitehead's question in the 1950s was about finding a purely algebraic test for exactly this property, instead of writing down a basis by hand.

F \cong \bigoplus_{i \in I} \mathbb{Z}

Detailed analysis

An abelian group $F$ is free if $F\cong\bigoplus_{i\in I}\mathbb{Z}$ for some basis $I$ : every element is a unique finite $\mathbb{Z}$ -linear combination of basis elements, with no relations except commutativity. Freeness is projective in the category of abelian groups — homomorphisms out of $F$ lift through any surjection — which is exactly why it matters: it is the property that makes resolutions split and homological invariants vanish.

J.H.C. Whitehead encountered the question in the 1950s through homotopy theory, where cohomology computations on spaces reduce to abelian group extensions, and asked whether a purely homological condition could certify freeness without exhibiting a basis: does $\mathrm{Ext}^1(W,\mathbb{Z})=0$ imply $W$ free? The question appears in Cartan–Eilenberg's foundational 1956 text as an open problem.

The converse direction is automatic: if $W$ is free, every extension $0\to\mathbb{Z}\to G\to W\to0$ splits because each basis element of $W$ lifts independently to $G$ — no relations need preserving. The whole content of the problem is whether the homological test detects freeness, not merely is implied by it. The next step defines the test precisely.

Terms in this step

Abelian group: A set with a commutative addition operation ( $a+b=b+a$ ), an identity element $0$ , and inverses for every element — the basic algebraic structure carried by $\mathbb{Z}$ , $\mathbb{Q}$ , and countless other familiar number systems.
Basis of a free abelian group: A subset of a free abelian group $F$ such that every element of $F$ is a unique finite integer combination of basis elements — the direct generalization of a vector space basis to abelian groups.

Knowledge used in this step

Group theory