Weak equivalences of pairs induce isomorphisms on relative homotopy

All source pairs below are finite CW pairs. Give each interval its two vertices and open edge, and each cube its product faces: a $d$-face has closure a closed $d$-cube, radially homeomorphic to a disk, with boundary its lower faces. Finite pasting gives the weak topology for this finite closed-face cover, so this is a finite CW structure. Unions of faces are subcomplexes. In particular $(S,J)$, $(D,S)$, and the cylinder face pairs used below meet [F3, F4]. This verification concerns only finite cubes, not products of arbitrary CW spaces.

To prove surjectivity let $u:D\to Y$ represent a relative class, so $u(S)\subseteq B$ and $u{\mathrm{\mid }}_J=b$. Apply [F3] to $g:A\to B$, the source pair $(S,J)$, target map $u{\mathrm{\mid }}_S$ and prescribed constant lift $a$ on $J$, with constant comparison there. Obtain $v:S\to A$ with $v{\mathrm{\mid }}_J=a$ and a homotopy $T:u{\mathrm{\mid }}_S\simeq gv$ in $B$ fixed on $J$. Apply [F4] to extend $T$, viewed in $Y$, to a homotopy $E:D\times I\to Y$ starting at $u$. It remains fixed on $J$ and sends $S$ into $B$ at every time, because those are its prescribed boundary values. Its endpoint $U$ satisfies $U{\mathrm{\mid }}_S=fv$.

To prove injectivity, take relative cubes $w_0,w_1:D\to X$ and a relative homotopy $H:D\times I\to Y$ from $f{w}_0$ to $f{w}_1$. Write $Q=D\times I$, $V=S\times I$, and $V_0=(S\times \{0,1\})\cup (J\times I)\mathrm{.}$ On $V_0$ prescribe a map $z:V_0\to A$ by $z(x,0)=w_0(x)$, $z(x,1)=w_1(x)$ for $x\in S$, and $z(x,t)=a$ for $x\in J$. These prescriptions agree at the intersections since both relative cubes are constant on $J$; finite closed pasting gives continuity. The restriction $H{\mathrm{\mid }}_V$ takes values in $B$, and $H{\mathrm{\mid }}_{V_0}=gz$.

Apply [F3] to $f:X\to Y$ and $(D,S)$ with target $U$, prescribed lift $v$ on $S$ and constant comparison $U{\mathrm{\mid }}_S=fv$. Obtain $w:D\to X$ extending $v$ and a homotopy $U\simeq fw$ rel $S$. The cube $w$ is relative: $w(F)\subseteq A$ and $w(J)=a$. Concatenation with $E$ gives a relative homotopy $u\simeq fw$ fixed on $J$. Hence every target relative class is in the image, in degree one as well as higher degrees.

Use [F3] for $g$ on the finite pair $(V,V_0)$, target $H{\mathrm{\mid }}_V$, prescribed lift $z$, and constant comparison on $V_0$. It yields $v:V\to A$ extending $z$ and a homotopy $T:H{\mathrm{\mid }}_V\simeq gv$ fixed on $V_0$. Let $R=\mathrm{\partial }Q=(D\times \{0,1\})\cup V\mathrm{.}$ On $R$ define a homotopy $E_R$ by $T$ on $V$ and by the stationary maps $f{w}_0,f{w}_1$ on the two end cubes. They agree on $S\times \{0,1\}$ because $T$ is fixed there. Thus $E_R$ is continuous, starts at $H{\mathrm{\mid }}_R$, fixes both end cubes, and fixes $J\times I$ at $b$.

Apply [F4] for $(Q,R)$ to extend $E_R$ to a homotopy in $Y$ starting at $H:Q\to Y$. Write $U:Q\to Y$ for its endpoint. The maps $w_0,w_1$ on the two end cubes and $v$ on $V$ glue to a continuous $z_R:R\to X$, since $v$ extends the endpoint data $z$. The endpoint boundary equation is $U{\mathrm{\mid }}_R=f{z}_R$. Apply [F3] to $f$ on $(Q,R)$ with this prescribed lift and constant comparison. Obtain $W:Q\to X$ with $W{\mathrm{\mid }}_R=z_R$. Therefore $W(-,0)=w_0$, $W(-,1)=w_1$, $W(F\times I)\subseteq A$, and $W(J\times I)=a$. Thus $W$ is the required relative homotopy, proving injectivity by equality of arbitrary classes, not merely by testing the distinguished class.

Steps 3.1 and 4.1 give bijectivity in every positive degree. By [F2] the induced map is pointed in every degree and is a homomorphism for $n\ge 2$, so its bijectivity makes it a group isomorphism in that range. The point $a$ was arbitrary, and no point or path was selected for a family of basepoints.

For $n=1$, $F=\{0\}$ and $J=\{1\}$; $(S,J)$ adds just one vertex, and $(V,V_0)$ adds the initial-endpoint interval while the terminal-endpoint interval stays constant. The constructions therefore apply literally to relative paths with their variable initial point in $A$. They require no group structure on relative ${\pi }_1$. Relative degree zero is not asserted. If $A$ is empty there is no $a$, and the quantified conclusion is vacuous; no source cube at a nonexistent basepoint is requested. Equal pairs, constant cubes and coincident endpoint maps cause no change. All homotopies fix the stated faces at every time, including their corners and endpoints; the finite HELP time reparametrization preserves each stationary prescribed track. All calls to [F3] have finite sources, and [F4] is choice-free. This proves the assertion without AC.