Relative homotopy compares with the CW quotient in the connectivity range

Apply [F2] to the constant cellular map $A\to \{v\}$, with empty fixed subcomplex. Its ordinary cylinder, after reversing the interval coordinate, is exactly the cone $CA$ of [F3], with $A$ as its free-end subcomplex and $v$ as apex. It is CW, and its retraction to $v$ induces a component bijection and isomorphisms on all positive groups at every basepoint. In particular $CA$ is path-connected and has trivial positive homotopy groups. Its explicit contraction is $[x,u]\mapsto [x,u+t(1-u)]$, from the identity to the apex; quotient-times-interval continuity is included in [F2].

Build $Y=X{\cup }_{A}CA$ from $X$ by adjoining the apex vertex and then the remaining cells of $CA\setminus A$. Their boundary maps have finite support and are cellular, so [F2] gives a CW complex containing $X$ and $CA$ as subcomplexes with intersection exactly $A$. Its map-out test is continuity on $X$ and $CA$ with agreement on $A$, since these tests are precisely their supplied characteristic-disk tests. Thus this is the ordinary amalgamated union, not a different topology on that set.

The pair $(CA,A)$ is $(s+1)$-connected. Component-surjectivity holds because $CA$ is path-connected and $A$ nonempty. In the pointed tail of [F4], ${\pi }_0(A)\to {\pi }_0(CA)$ is bijective since both spaces are path-connected. Thus every relative degree-one class is in the image of ${\pi }_1(CA)$, which is zero by step 1.1. For $2\le j\le s+1$, the adjacent absolute cone groups are zero, and [F4] identifies ${\pi }_j(CA,A,a)$ with ${\pi }_{j-1}(A,a)$; the latter is zero by $s$-connectedness. This range is empty for $s=0$. These calculations hold at the specified arbitrary point $a$ and also at every other point of $A$.

Apply [F5] to the union in step 2.1, whose common subcomplex $A$ is nonempty path-connected. Its two pair connectivities are $r$ for $(X,A)$ and $s+1$ for $(CA,A)$. We obtain $e_{\ast }:{\pi }_i(X,A,a)⟶{\pi }_i(Y,CA,a)$ bijective for $1\le i<r+s+1$ and surjective for $i=r+s+1$. All the hypotheses, including the endpoint when $r=s=0$, are covered by steps 1.1–2.2.

Collapse $CA$ inside $Y$. It is a nonempty contractible subcomplex by steps 1.1–2.1, so [F6] makes $p:Y\to Y\mathrm{/}CA$ a weak equivalence. The restriction $CA\to \{\ast \}$ is also weak by step 1.1. Hence the pair comparison of [F6] induces a bijection $p_{\ast }:{\pi }_i(Y,CA,a)⟶{\pi }_i(Y\mathrm{/}CA,\ast ,\ast )$ in every positive degree. The relative target classes are precisely absolute based cubes by [F1]; no nontrivial boundary values remain. For $i\ge 2$ the comparison preserves the group operations, while at $i=1$ this is an identification of the underlying pointed sets.

There is a canonical homeomorphism $Y\mathrm{/}CA\cong X\mathrm{/}A$. Set-theoretically it retains exactly the points of $X\setminus A$ and the one collapsed point. A function out of $Y\mathrm{/}CA$ is continuous exactly when its composite on $Y$ is continuous and constant on $CA$. By the union map-out test in step 2.1, this says exactly that its restriction on $X$ is continuous and constant on $A$, which is the quotient map-out criterion for $X\mathrm{/}A$. Testing characteristic maps into the two-point open-set classifier, as in [F6], proves equality of the two topologies. Under this identification, $p{\mathrm{\mid }}_X$ is the original quotient map $q:X\to X\mathrm{/}A$. Consequently $p_{\ast }{e}_{\ast }=q_{\ast }$ on the cubical relative representatives. Combining steps 3.1 and 3.2 gives bijectivity for the integral indices $1\le i\le r+s$ and surjectivity at $r+s+1$, as claimed.

For $r=s=0$ only the positive degree-one surjection is asserted, and step 3.1 gives it; no relative degree-zero group has been introduced. If $A=X$, both the relative source sets and the positive groups of the one-point quotient are trivial, consistent with every claimed range. A singleton $A$ and no relative cells are also allowed. Every cone endpoint and quotient value is fixed by its defining relation; the argument uses the arbitrary original basepoint $a$, which need not be a vertex. The cone contraction is explicit, its collapse uses choice-free HEP, and the excision theorem and relative weak comparison are choice-free. Therefore this entire comparison requires no AC, including for infinite CW complexes.