Relative homology and stabilization of CW-skeleta

Proof. For part (a), the case \(n=0\) follows because \(X^0\) is a disjoint union of points. For \(n\geq 1\), the pushout square defining \(X^n\) from \(X^{n-1}\) shows that \[ X^n/X^{n-1} \;\simeq \; \bigvee _{\alpha \in I_n} D^n/S^{n-1} \;\simeq \; \bigvee _{\alpha \in I_n} S^n, \] a wedge of \(n\)-spheres indexed by the \(n\)-cells. Therefore \[ H_k(X^n, X^{n-1}) \;=\; \widetilde {H}_k\Big (\bigvee _{\alpha \in I_n} S^n\Big ) \;\cong \; \bigoplus _{\alpha \in I_n} \widetilde {H}_k(S^n) \;\cong \; \begin {cases} \bigoplus _{\alpha \in I_n} \Z & \text {if } k = n, \\ 0 & \text {if } k \neq n. \end {cases} \]

For part (b), the claim is clear for \(n = 0\): we have \(H_k(X^0) = 0\) for \(k > 0\) as \(X^0\) is a disjoint union of points, which have trivial homology. Consider now the long exact sequence of the pair \((X^n, X^{n-1})\): \[ H_{k+1}(X^n, X^{n-1}) \to H_k(X^{n-1}) \to H_k(X^n) \to H_k(X^n, X^{n-1}). \] By part (a), the first term vanishes when \(k \neq n - 1\) and the last term vanishes when \(k \neq n\). In particular, for \(k > n\) the map \(H_k(X^{n-1}) \to H_k(X^n)\) is an isomorphism and the claim follows by induction. For part (c), we may similarly use the long exact sequence of the pair \((X^{n+1},X^n)\) to conclude that the map \(H_k(X^n) \to H_k(X^{n+1})\) is surjective for \(k \leq n\) and injective (hence an isomorphism) for \(k < n\).

It remains to pass from the skeleta to their colimit. Adding a disjoint basepoint commutes with colimits. Apply the pushout presentation of Lemma 2.4.29 to the sequence \((X^r)_+\). Since the resulting square is a pushout, the cofiber of its top horizontal map is isomorphic to the cofiber of its bottom horizontal map. Comparing the corresponding long exact sequences gives the Mayer–Vietoris sequence of the pushout. By the wedge axiom, and after multiplying the odd source summands by \(-1\), the relevant portion is \[ \bigoplus _{r\geq 0}H_k(X^r) \xrightarrow {\delta _k} \bigoplus _{r\geq 0}H_k(X^r) \longrightarrow H_k(X) \longrightarrow \bigoplus _{r\geq 0}H_{k-1}(X^r) \xrightarrow {\delta _{k-1}} \bigoplus _{r\geq 0}H_{k-1}(X^r). \] Writing \(\iota _r\) for the inclusion of the \(r\)-th summand, the first map is given by \[ \delta _k(\iota _r(x))=\iota _r(x)-\iota _{r+1}((X^r\to X^{r+1})_*(x)). \] The map \(\delta _k\) is injective: if an element of the direct sum lies in its kernel, its components vanish successively, starting with the zeroth component. Its cokernel is the algebraic colimit \(\colim _rH_k(X^r)\). The same injectivity statement for \(\delta _{k-1}\) therefore identifies the middle map with an isomorphism \(\colim _rH_k(X^r)\iso H_k(X)\). The claim now follows from the stabilization properties established above. □