Prove the following uniqueness result for the Quillen plus construction: Given a connected CW complex \(X\), if there is an abelian CW complex \(Y\) and a map \(X \to Y\) inducing an isomorphism \(H_*(X; \mathbb{Z}) \approx H_*(Y; \mathbb{Z})\), then such a \(Y\) is unique up to homotopy equivalence.
Solution
Suppose \[ i:X\to Y, \qquad i':X\to Y' \] are homology isomorphisms and \(Y,Y'\) are connected abelian CW complexes. Replace \(i\) by the inclusion of \(X\) into its mapping cylinder \(W=M_i\). Then \(W\simeq Y\), and since \(i\) is a homology isomorphism, \[ H_*(W,X;\mathbb Z)=0. \] The universal coefficient theorem therefore gives, for every abelian group \(A\), \[ H^{n+1}(W,X;A)=0 \qquad\text{for all }n. \] In particular this holds for \[ A=\pi_n(Y'). \]
Because \(Y'\) is an abelian CW complex, Corollary 4.73 applies: all obstruction groups to extending \[ i':X\to Y' \] over \(W\) vanish. Hence there is a map \[ F:W\to Y' \] extending \(i'\). Restricting along the deformation retract \(Y\simeq W\) gives a map \[ f:Y\to Y' \] with \[ fi\simeq i'. \] Since both \(i\) and \(i'\) induce homology isomorphisms, so does \(f\). Proposition 4.74, the homology Whitehead theorem for connected abelian CW complexes, therefore gives \[ \boxed{f:Y\xrightarrow{\simeq}Y'.} \] Thus the abelian target of the Quillen plus construction is unique up to homotopy equivalence.