Relative homology of consecutive CW skeleta

For $n\ge 1$, the pair $(X^n,X^{n-1})$ is a good pair (collar neighborhoods in the attached disks retract to $X^{n-1}$), and collapsing $X^{n-1}$ gives the wedge ${\bigvee }_{\alpha }{D}^n\mathrm{/}{S}^{n-1}\cong {\bigvee }_{\alpha }{S}^n$. For $n=0$, $X^0$ is discrete and the same direct-sum calculation follows directly from its singular chain complex.

If there are no $n$-cells, both sides are zero. Otherwise, for $n\ge 1$, Good pairs and quotient reduced homology and Homology of spheres give one copy of $G$ in degree $n$ and zero otherwise. A singular simplex has compact image, so A compact subspace of a CW complex meets only finitely many cells makes every chain involve only finitely many wedge summands; the wedge group is therefore the direct sum, not a product. The direct $n=0$ calculation in step 1.1 gives the same conclusion.