Proof of Theorem 14.2. We first reduce to CW complexes. In addition to
Theorem 9.17, we use the standard facts that weak homotopy equivalences preserve singular homology and that CW approximation has a relative version (see Hatcher, § 4.1). Approximating both spaces of a pair and comparing their long exact sequences preserves relative homotopy and homology as well. We may therefore assume that
\(X\) is a CW complex and
\((X,A)\) is a CW pair.
Reduction of (ii) to (i). Apply
Lemma 13.6 with
\(r=n-1\) and
\(s=1\text{.}\) Since
\((X,A)\) is
\((n-1)\)-connected and
\(A\) is simply connected, it gives
\(\pi_i(X, A) \cong \pi_i(X/A)\) for
\(i \leq n\text{.}\) Singular homology of pairs satisfies
\(H_i(X, A) \cong \widetilde{H}_i(X/A)\) for all
\(i\) (a standard CW result). The homotopy comparison shows that
\(X/A\) is
\((n-1)\)-connected, so (i) applied to
\(X/A\) gives (ii).
Proof of (i). The refined CW approximation for an \((n-1)\)-connected space gives a model with one \(0\)-cell and no cells in dimensions \(1,\ldots,n-1\) (see Hatcher, § 4.1). Its cellular chain complex immediately gives \(\widetilde H_i(X)=0\) for \(i<n\text{.}\) Both \(\pi_n\) and \(H_n\) are determined by the \((n+1)\)-skeleton, so we may also discard all higher cells. We are left with
\begin{equation*}
X = \bigvee_\alpha S^n_\alpha \cup_{\varphi_\beta} \bigcup_\beta e^{n+1}_\beta.
\end{equation*}
\begin{equation*}
\pi_n(X) \cong \biggl(\bigoplus_\alpha \mathbb{Z}\biggr) \big/ \bigl\langle [\varphi_\beta] \bigr\rangle.
\end{equation*}
This presentation matches the cellular boundary
\begin{equation*}
d\colon H_{n+1}(X^{n+1},X^n)\longrightarrow H_n(X^n,X^{n-1}).
\end{equation*}
For each cell \(e^{n+1}_\beta\text{,}\) the coefficient of \(e^n_\alpha\) in \(d e^{n+1}_\beta\) is the degree of \(q_\alpha \circ \varphi_\beta\text{,}\) where \(q_\alpha\) collapses every \(n\)-cell except \(e^n_\alpha\) to a point — precisely the \(\alpha\)-component of \([\varphi_\beta] \in \bigoplus_\alpha \mathbb{Z}\text{.}\) There are no \((n-1)\)-cells, so the outgoing cellular differential from degree \(n\) is zero. Every \(n\)-chain is therefore a cycle, and the boundaries are exactly the image of \(d\text{.}\) Thus homotopy and homology have the same generators and relations:
\begin{equation*}
\pi_n(X)\cong\operatorname{coker}(d)\cong\widetilde H_n(X).
\end{equation*}
With compatible orientations, the isomorphism sends each sphere inclusion to its homology class. It is therefore the Hurewicz map. For the relative case, the quotient map induces the commutative square
Both horizontal maps are the quotient comparisons used above. The square commutes because pushing forward a representative and then passing to the quotient gives the same homology class as doing these operations in the opposite order. Thus the relative isomorphism is also induced by the Hurewicz map.
