Relative Hurewicz theorem in the simple-connectivity range

All hypotheses of [F2] hold: the pair is CW and $(n-1)$-connected, $A$ is nonempty and simply connected, and AC is available. Thus there is a homotopy equivalent pair $(Z,A)$ rel $A$ with no relative cells below $n$. The lower homology assertion in [F2] gives $H_i(Z,A)=0$ for $0\le i<n$. The equivalence and inverse homotopies of pairs identify these groups with $H_i(X,A)$, as verified by the relative prism calculation in [F1]. Hence $H_i(X,A)=0$ throughout the required range, including degree zero.

Let $F_n$ and $F_{n+1}$ be the free abelian groups on the model's relative cells in those dimensions, and $D:F_{n+1}\to F_n$ its degree-incidence map. By [F2], both ${\pi }_n(Z,A,x_0)$ and $H_n(Z,A)$ are identified with $F_n\mathrm{/}\mathrm{im}D$, with $h$ induced by the identity of $F_n$. Explicitly, every homology class has a finite cell-vector representative $v\in F_n$, and the homotopy class represented by the same vector maps to it, proving surjectivity. If a homotopy class represented by $v$ has zero image, then $v\in \mathrm{im}D$ in the homology presentation. The homotopy presentation has exactly that same relation subgroup, so its class is zero, proving injectivity. Both maps are homomorphisms by [F1] and the presentations in [F2]. Naturality in [F1] and the equivalence rel $A$ transfer this isomorphism to the displayed $h$ on $(X,A,x_0)$; the equivalence fixes $x_0$, so no basepoint change is concealed.

When $n=2$, [F2] proves that the first relative cell group is free abelian under simple connectivity of $A$ and that its surjective image in the full relative group has the identical cokernel presentation. Thus that full group is abelian before identifying it with homology. The result does not replace a potentially nonabelian group by its abelianization without justification. An equal pair gives zero groups; no relative cells give zero free groups, and a single cell or a point subspace is covered by the same presentation. A based pair cannot have empty $A$; degree one is outside this relative assertion. The two kernel/image directions were established separately in step 2.1, with zero vectors included. AC is propagated exactly from the model-equivalence construction in [F2], as stated in [A1]. Dropping simple connectivity would invalidate that supplier's free-basis hypothesis, so this proof makes no assertion in that case. This completes the theorem.