Relative cellular homology computes relative singular homology

If $A=\mathrm{\varnothing }$, the assertion is the absolute cellular-homology theorem. Suppose $A\mathrm{\ne }\mathrm{\varnothing }$. The quotient $X\mathrm{/}A$ is a CW complex with one base vertex $\ast$ coming from $A$ and exactly the cells of $X\setminus A$ otherwise. In degree zero the reduced cellular group is the augmentation kernel in $C_0^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(X\mathrm{/}A;G)$. It is identified with the quotient degree-zero group by sending each outside vertex generator $e$ with coefficient $g$ to $g(e-\ast )$; this is an isomorphism even when there are infinitely many vertices, since each chain has finite support. In higher degrees the quotient and reduced groups have identical outside-cell bases. Their differentials agree: deleting an $A$-vertex in the quotient is exactly rewriting it as the base vertex in the augmentation-kernel basis. Thus the two complexes are canonically isomorphic in every degree.

The published CW-quotient comparison A CW quotient induces relative singular homology isomorphisms directly identifies $H_{\ast }(X,A;G)$ with $H_{\ast }(X\mathrm{/}A,\{\ast \};G)={\tilde{H}}_{\ast }(X\mathrm{/}A;G)$ for nonempty $A$, without assuming a pre-existing neighborhood retraction. Applying Cellular homology computes singular homology to $X\mathrm{/}A$ and using step 1.1 gives the claimed groups. Under this identification, the triple connecting maps for the relative skeleta are exactly the quotient cellular differential.