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I've been working on problem 4.1.16 of Hatcher's Algebraic Topology and am at a complete impasse. The problem is as follows:

Show that a map $f:X→Y$ between connected CW complexes factors as a composition $X→Z_n→Y$ where the first map induces isomorphisms on $π_n$ for $i≤n$ and the second map induces isomorphisms on $π_n$ for $i≥n+1.$

I applied Proposition 4.13 on the pair $(M_f,X)$ to get an $n$-connected CW model $(Z_n,X)$. Because $M_f$ deformation retracts to $Y$, this gives the desired isomorphisms for the second map. Moreover, because $(Z_n,X)$ is $n$-connected, the inclusion of $X$ in $Z_n$ gives the desired isomorphisms for the first map, with the exception of $π_n(X)→π_n(Z_n)$ (this map is, however, surjective). How do I prove injectivity? Any help would be appreciated.

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  • $\begingroup$ I've come to the conclusion that the approach I was trying before (that is, using $(M_f,X)$) is wrong; by definition of an n-connected CW model, $\pi_n(Z_n) \to \pi_n(M_f)$ is injective, so if $\pi_n(X) \to \pi_n(Z_n)$ is injective, then the composition $\pi_n(X) \to \pi_n(M_f)$ is injective, so there exists an injective map from $\pi_n(X) \to \pi_n(Y)$. It is not too hard to find a counterexample to this statement. $\endgroup$ – Andrew Mar 6 '14 at 3:40
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I don't think you need the mapping cylinder or the CW model for this.

Start with the factorization $$ X \xrightarrow{i} Z_0 \xrightarrow{g} Y, $$

where $Z_0 = X$, $i = \Bbb 1_X$ and $g = f$. Obviously $i_*$ is an isomorphism on $\pi_j$ for $j \le n$. We will attach $k$-cells to $Z_0$ for $k \ge n + 1$ and extend $g$ to make $g_*$ an isomorphism on $\pi_j$ for $j \ge n + 1$, while keeping $i_*$ an isomorphism for $j \le n$. We will use $Z_k$ to refer to the result of attaching $k$-cells to $Z_{k-1}$.

Fix base points for $Z_0$ and $Y$.

We will start by making $g_*$ surjective on $\pi_{n+1}$. Just as in the construction of CW approximations, choose maps $\varphi_\alpha : S^{n+1} \to Y$ representing the generators of the group $\pi_{n+1}(Y)$. For each $\varphi_\alpha$, attach an $(n+1)$-cell via a constant map to the base point of $Z_0$. This gives us $Z_{n+1}$. Extend $g$ to $Z_{n+1}$ via the maps $\varphi_\alpha$. The resulting $g_*$ is surjective on $\pi_{n+1}$ by construction.

Since the pair $(Z_{n+1}, X)$ is $n$-connected, $i_*$ is still an isomorphism for $j < n$, and surjective for $j = n$. Since $Z_{n+1}$ is the wedge sum of $X$ with $(n+1)$-spheres, there is a retraction from $Z_{n+1}$ onto $X$. This makes $i_*$ injective on $\pi_j$ for all $j$. Thus, $i_*$ is still an isomorphism on $\pi_j$ for $j \le n$.

Now we will attach $k$-cells to $Z_{n+1}$ for $k > n + 1$ to make $g_*$ an isomorphism on $\pi_j$ for $j \ge n + 1$. Such cells won't affect homotopy groups for $j \le n$, and therefore $i_*$ will remain an isomorphism in these dimensions.

For each $k > n + 1$, inductively attach $k$-cells to $Z_{k-1}$ to make $g_*$ injective on $\pi_{k-1}$ and surjective on $\pi_k$, exactly as done in the construction of CW approximations. The end result is the desired factorization.

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I think a possible solution is to let $X_n$ be the $n$th space in the Postnikov tower of $X$, and let $Y^n$ be an $n$-connected CW model for $(Y, y_0)$, as constructed in Proposition 4.13. Let $Z_n$ be the disjoint union of $X_n$ and $Y^n$. Let $X \to Z_n$ be the inclusion of $X$ in $X_n$ and $Z_n \to Y$ be the disjoint union of $Y^n \to Y$ (from the CW model) and the map taking all of $X_n$ to $y_0$. Does this solution appear to be correct?

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  • $\begingroup$ This doesn't work. It's not a factorization of $f$. See the answer I added. $\endgroup$ – Ayman Hourieh Mar 14 '14 at 21:03

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