A question about the proof that a homotopy equivalence induces an isomorphism

If $f:X \rightarrow Y$ is a homotopy equivalence with homotopy inverse $g$ and I want to prove that $f_*$ is an isomorphism then Hatcher (on page 37) uses the following:

$$\pi_1 (X, x_0) \xrightarrow{f_*} \pi_1 (Y, f(x_0)) \xrightarrow{g_*} \pi_1 (X, g(f(x_0))) \xrightarrow{f_*} \pi_1 (X, f(g(f(x_0))))$$

My question is: why is it useful or necessary to apply the functions several times? If I have $gf \simeq id_X$ then I know $(gf)_* = \beta_h$ (preceding lemma) where $\beta_h$ is an isomorphism so $f_*$ is injective. Doing the same for $fg \simeq id_Y$ yields $f_*$ is surjective, so I'm done.

What am I missing? Thanks for your help!

-

1 Answer

You're missing nothing. And Hatcher is doing the same thing, but just draws both $(fg)_*$ and $(gf)_*$ on the same diagram.

-