Relative homology of consecutive CW skeleta

If there are no $n$-cells, $X^n=X^{n-1}$ and both sides are zero. For $n=0$, $X^0$ is discrete and $X^{-1}=\mathrm{\varnothing }$. Every singular simplex has constant image, so its singular chain complex splits over the vertices. On one point its degree-$k$ chain group is $G$ and its boundary is multiplication by ${\sum }_{i=0}^k(-1{)}^i$, which is zero for odd $k$ and the identity for positive even $k$; hence its homology is $G$ in degree zero and zero otherwise. This gives the asserted direct sum without an additivity axiom. Hence assume $n\ge 1$ and that an $n$-cell exists. Its attaching sphere makes $X^{n-1}$ nonempty.

We verify the good-pair hypothesis for $(X^n,X^{n-1})$. For each $n$-cell use its supplied characteristic map ${\chi }_e:D^n\to X^n$, and let $V=X^{n-1}\cup {\bigcup }_e{\chi }_e\{u\in D^n:\mathrm{\parallel }u\mathrm{\parallel }>1\mathrm{/}2\}\mathrm{.}$ The preimage of $V$ in each characteristic disk of dimension below $n$ is the whole disk, and in each $n$-disk it is the relatively open outer annulus $\{\mathrm{\parallel }u\mathrm{\parallel }>1\mathrm{/}2\}$; the CW weak topology therefore makes $V$ open. On each such annulus set $H_t({\chi }_e(u))={\chi }_e((1-t)u+tu\mathrm{/}\mathrm{\parallel }u\mathrm{\parallel }),$ and fix $X^{n-1}$ pointwise. The formula is continuous on each characteristic-disk cylinder and fixes the boundary sphere where $\mathrm{\parallel }u\mathrm{\parallel }=1$, so it agrees under every attaching identification, even for nonregular cells. Here is the product-continuity check for arbitrary cell sets. For each point of $V$, its track is continuous; transpose these tracks to a function $\widehat{H}:V\to C(I,V)$ with the compact-open topology. By The exponential law: for a locally compact metric $X$ and any spaces $Z$ and $Y$, transposition is a bijection between $C(X\times Z,Y)$ and $C(Z,C(X,Y))$ with the compact-open topology, the restriction of $\widehat{H}$ to every characteristic disk's inverse image of $V$ is continuous. The restricted characteristic maps jointly give the weak topology on the open subspace $V$: a subset of $V$ is open exactly when its inverse image in each disk is open. Hence $\widehat{H}$ is continuous, and the same interval exponential law untransposes it to a continuous $H:V\times I\to V$. At $t=1$ it lands in $X^{n-1}$; thus $V$ deformation retracts onto that closed subcomplex. By [F3], $H_k(X^n,X^{n-1};G)\cong {\tilde{H}}_k(W;G),W=X^n\mathrm{/}{X}^{n-1}={\bigvee }_e(D_e^n\mathrm{/}{S}_e^{n-1})\mathrm{.}$ The quotient is a wedge of $n$-spheres with common basepoint $b$ by the attaching presentation, and the isomorphism is induced by the quotient map.

First let $W_J$ be a finite wedge and split off one sphere $S_j^n$, leaving the subwedge $U$ that contains the common basepoint. This is again a skeletal attachment pair: the open neighborhood $U\cup {\chi }_j\{\mathrm{\parallel }u\mathrm{\parallel }>1\mathrm{/}2\}$ and the radial homotopy of step 1.2 retract onto $U$. Therefore [F3] applies, and the quotient $W_J\mathrm{/}U\cong S_j^n$ gives $H_k(W_J,U;G)\cong {\tilde{H}}_k(S_j^n;G)$. The inclusion $U\hookrightarrow W_J$ has a retraction collapsing $S_j^n$; hence its homology map is injective. In the pair long exact sequence this also forces the connecting map into $H_{k-1}(U;G)$ to be zero. Thus, for $k>0$, the sequence from $H_k(U;G)$ to $H_k(W_J;G)$ to ${\tilde{H}}_k(S_j^n;G)$ is short exact. It splits by the inclusion of $S_j^n$, since projection to the quotient is the identity on that sphere. Induction over the finite set $J$, together with sphere homology, gives $H_k(W_J;G)={⨁}_{j\in J}G$ for $k=n$ and zero for other positive $k$. Every finite wedge is path connected, including the empty wedge interpreted as a point, so ${\tilde{H}}_0(W_J;G)=0$.

The choice-free compact-image lemma puts the image of each singular simplex in the arbitrary wedge $W$ inside a finite CW subcomplex, hence inside a finite subwedge. Every singular chain is a finite sum of simplices, so it too is supported in a finite subwedge. A cycle in $W$ supported there is a cycle there, since singular chain inclusion is injective. If a finite supported cycle bounds in $W$, its bounding chain also lies in some finite subwedge; enlarging both finite supports makes it a boundary in one finite subwedge. Therefore the finite-wedge calculations of step 2.1 pass elementwise to $W$: classes have finite support, and any relation is already a finite-wedge relation. This gives ${\tilde{H}}_n(W;G)\cong {⨁}_{e_{\alpha }^n}G$ and zero in every other degree, including reduced degree zero because $n\ge 1$ and $W$ is path connected. Composing with the quotient-induced isomorphism of step 1.2 proves the stated natural isomorphism, without a choice premise.