So I am currently trying to understand the given proof of Hatcher's proof of proposition 4.13.
It's this particular part (in the middle of the screenshot) I don't understand:
The extended $f$ still induces a surjection on $\pi_k$ since the composition $\pi_k(Z_k)\to \pi_k(Y_{k+1}) \to \pi_k(X)$ is surjective.
How do we know that this composition is surjective? I understand the implication, but I don't understand why the composition is supposed to be surjective in the first place.
Does that follow from the induction hypothesis, that $\pi_k(Z_k) \to \pi_k(X)$ is surjective? If so, why does "factorizing through $\pi_k(Z_{k+1})$" not affect surjectivity?
I am sorry for not being able to express my confussion in any better way.
I hope someone can enlighten me.
Thank you very much!
If $\pi_k(Z_k) \to \pi_k(X)$ is surjective, well, the composite $\pi_k(Z_k) \to \pi_k(Y_{k+1}) \to \pi_k(X)$ is that same map, so it is surjective. So yes, it follows from the inductive hypothesis.