How to get the above statement as a corollary of the relative Hurewicz theorem (in the version below)?

Let $(X,A)$ be an n-connected topological pair with $n\geq 2$. Then the Hurewicz map $\pi_n(X, A)/N \rightarrow H_n(X,A)$ is an isomorphism, where $N$ is the normal subrgoup generated by elements of the form $(\gamma f)-f$ with $\gamma\in\pi_1(A)$ and $f\in\pi_n(X,A)$.

A set of lecture notes I've been going through proves Hurewicz first and cellular approximation later. This statement is used in the proof and the only comment is that Hurewicz reduces that to basically excision in homology. I have two problems here:

  1. To even think about using Hurewicz, I need some proof that $(X, X^{(n)})$ is 1-connected anyway.
  2. Even if the assumption is satisfied all we get is that $\pi_n(X, X^{(n)})\subset N$ which, I gather, can be nontrivial.

The following is taken from Topology and Groupoids.

7.6.1 The following statements are true for each $n \geqslant 1$.

$\alpha(n)$ Any map $S^r \to S^n$ with $r < n$ is inessential.

$\beta(n)$ Any map $S^r \to S^n$ with $r < n$ extends over $E^{r+1}$.

$\gamma(n)$ Let $B$ be path-connected and let $Q$ be formed by attaching a finite number of $n$-cells to $B$. Then any map $$ (E^r,S^{r-1}) \to (Q,B)$$ with $r<n$ is deformable into $B$.

The proof is by induction by means of the implications $$ \gamma(n) \Rightarrow \alpha(n) \Leftrightarrow \beta(n) \Rightarrow \gamma(n+1)$$ the only difficult step being the proof of $\beta(n) \Rightarrow \gamma(n+1)$. The start of the induction---the proof of $\gamma(1)$---is easy; in fact, since $E^0$ consists of a single point and $S^{-1}$ is the empty set, $\gamma(1)$ is equivalent to the statement that $Q$ is path-connected [which is easy to prove].

The result is taken from very old notes on homotopy theory of J.F. Adams, and uses subdivision and deformation arguments. The Cellular Approximation Theorem is a consequence of the result.

I'm not sure if this answers your titled question.

A quite different approach to these questions is taken in the book Nonabelian Algebraic Topology where the Relative Hurewicz Theorem is deduced from a Higher Homotopy Seifert-van Kampen Theorem. This takes a lot of setting up of the necessary background algebra and homotopical constructions, but does not involve the usual homology theory.


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