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.
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?