A relative single cell layer has compatible homotopy and homology bases

The quotient cell construction [F4] identifies $Z\mathrm{/}A$ with the CW wedge $W={\bigvee }_e{S}_e^k$: every remaining boundary is sent to the quotient vertex and each open cell is unchanged. For every based space $V$ and positive $k$, the pair sequence in [F6] makes $H_k(V)\to H_k(V,\ast )$ an isomorphism: the adjacent positive homology groups of the point vanish, and in degree one the map $H_0(\ast )\to H_0(V)$ is injective because the map $V\to \ast$ supplies a left inverse. Orient $S_e^k=D^k\mathrm{/}{S}^{k-1}$ by the image of $[D^k,S^{k-1}]$ under disk quotient followed by the inverse of this point-relative isomorphism. This image is a generator, because [F4] applies also to the standard finite CW pair $(D^k,S^{k-1})$ and gives a quotient-induced homology isomorphism. Thus the orientation convention is specified, not an unspecified possible sign. Denote the inclusion of this sphere into $W$ by ${\iota }_e$.

By [F1], $(Z,A)$ is $(k-1)$-connected. Since $A$ is simply connected it is $1$-connected. Apply [F2] with $r=k-1$, $s=1$: $q_{\ast }:{\pi }_k(Z,A,a)⟶{\pi }_k(W,\ast )$ is an isomorphism, including at the endpoint $k=r+s$. By [F3], the target is free abelian on the classes $[{\iota }_e]$. Define $c_e$ to be their unique inverse images under $q_{\ast }$. These inverses exist individually and are unique, hence define the whole family without AC. In particular the relative degree-two group here is abelian; it is not merely presumed abelian for an arbitrary pair.

Fix one cell and a marked point $v\in S^{k-1}$. There is a path $\gamma$ in $A$ from ${\chi }_e(v)$ to $a$. Use a finite CW structure on the boundary sphere with $v$ as vertex, for example its one-vertex and one-top-cell structure. HEP [F5] extends the homotopy $v\mapsto \gamma (t)$ from that vertex to a homotopy of ${\chi }_e{\mathrm{\mid }}_{S^{k-1}}$ in $A$. Apply HEP again to $(D^k,S^{k-1})$ with target $Z$ to extend this boundary homotopy and the initial map ${\chi }_e$ over the disk. Its final map ${\chi }_e^{\mathrm{\prime }}$ has its whole boundary in $A$ and sends $v$ to $a$, so is a based relative disk representative. The homotopy remains a homotopy of pairs, although its marked value moves. After applying $q$, its entire boundary is constantly the quotient vertex at every time. It therefore descends to a based homotopy of quotient spheres: quotient-times-interval continuity follows from the explicit HEP proof [F5]. The initial quotient sphere map is ${\iota }_e$, so $q_{\ast }[{\chi }_e^{\mathrm{\prime }}]=[{\iota }_e]$. By the injectivity in step 1.2, $[{\chi }_e^{\mathrm{\prime }}]=c_e$, independently of the path and extensions. Only finitely many witnesses for this one cell were used; no family of paths or extensions was selected.

By [F4] and the point-relative comparison proved in step 1.1, the composite $H_k(Z,A)\stackrel{q_{\ast }}{\to }{H}_k(W,\{\ast \})⟶H_k(W)$ is an isomorphism. On $u_e=({\chi }_e{)}_{\ast }[D^k,S^{k-1}]$ it gives $({\iota }_e{)}_{\ast }[S_e^k]$, since quotient and characteristic maps commute pointwise and the sphere orientation was defined exactly in step 1.1. By the wedge homology calculation in [F4], these images form a free abelian basis. Hence the $u_e$ form a free abelian basis of $H_k(Z,A)$.

The homotopy of pairs in step 2.1 gives $({\chi }_e^{\mathrm{\prime }}{)}_{\ast }[D^k,S^{k-1}]=({\chi }_e{)}_{\ast }[D^k,S^{k-1}]$ by [F6]; its moving marked value does not obstruct the prism identity on relative chains. The definition of $h$ and step 2.1 therefore give $h(c_e)=u_e$. Additivity in [F6] now identifies the two free abelian groups on all finite sums, not just on the displayed generators. If there are no cells, $Z=A$, the relative groups and the empty free abelian group are zero. One cell gives one copy of $\mathbb{Z}$; the base space may be a point and the specified $a$ need not be a CW vertex. Degree zero and degree one are excluded; the degree-two case was explicitly justified by the quotient isomorphism. Cell orientations are supplied, quotient inverses are unique and the only discretionary witnesses were finite ones for a single cell. Thus no choice principle is used.