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I'm working on problem 16 in section 4.1 of Hatcher's Algebraic Topology book. I really have no ideas so far:

Show that a map $f: X \to Y$ between connected CW complexes factors as a composition $X \to Z_n \to Y$ where the first map induces isomorphisms on $\pi_n$ for $i\le n$ and the second map induces isomorphisms on $\pi_n$ for $i\ge n+1$.

Any help is appreciated.

Tony B
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1 Answers1

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Consider proposition 4.13 in Hatcher with respect to the pair $(M_f,X)$, where $M_f$ is the mapping cylinder of $f:X\to Y$. This should get you everything but the isomorphism $\pi_n(X)\to \pi_n(Z_n)$.

To handle this last case, take a look at the argument for constructing a Postnikov tower (example 4.17) for inspiration.

fixedp
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  • Unless I'm mistaken, you're saying that one should use Proposition 4.13 to obtain an n-connected CW model $(Z_n,X)$ for $(M_f,X)$. By definition of an n-connected CW model, the induced $\pi_n(Z_n) \to \pi_n(M_f)$ is an injection, so if the statement we are trying to prove is true, the composition $X \to Z_n \to M_f$ induces an injection $\pi_n(X) \to \pi_n(M_f)$, and, via the deformation retraction of $M_f$ onto $Y$, there exists an injection $\pi_n(X) \to \pi_n(Y)$. This is not true in general. – Andrew Mar 07 '14 at 06:10