Show that \([X, Y]\) is finite if \(X\) is a finite connected CW complex and \(\pi_i(Y)\) is finite for \(i \leq \dim X\).
Solution
Let \(d=\dim X\). Replace \(Y\) by its \(d\)-th Postnikov stage \(Y^d\). The map \[ Y\to Y^d \] induces isomorphisms on \(\pi_i\) for \(i\le d\), so obstruction theory (or cellular induction) gives a bijection \[ [X,Y]\cong[X,Y^d] \] for the \(d\)-dimensional CW complex \(X\).
We now construct maps \(X\to Y^d\) cell by cell. On the \(0\)-skeleton there is only one choice up to homotopy because \(X\) is connected and we may fix a basepoint component of \(Y\). Suppose a map has been constructed on \(X^{k-1}\). For each \(k\)-cell, extension across the cell is obstructed by an element of \[ \pi_{k-1}(Y), \] and, when extensions exist, the set of choices up to homotopy rel boundary is a torsor under \[ \pi_k(Y). \] Both groups are finite for \(k\le d\) by hypothesis.
Since \(X\) has only finitely many cells, at each stage there are only finitely many possible obstruction values and finitely many choices of extensions. Inducting over the finitely many skeleta, only finitely many homotopy classes of maps can occur. Thus \[ \boxed{[X,Y]\text{ is finite}.} \]