Let $(X,x_0)$ be a based retract of $(Y,y_0)$ with retraction $R$ and inclusion $\iota$ such that $$R\circ\iota=\mathrm{id}_X.$$

Show that there is a short exact sequence $$1\rightarrow N\rightarrow\pi_1(Y,y_0)\xrightarrow{R_*}\pi_1(X,x_0)\rightarrow 1,$$ where $N:=\ker(R_*)$.

So I thought it suffices to show that $R_*$ is surjective. We have $$R_*\circ\iota_*=\mathrm{id}_{\pi_1(X,x_0)}$$ But this already implies that $R_*$ is surjective and since $N=\ker(R_*)$, the above is an exact sequence.

This seems too simple to me, did I miss something?

  • 1
    $\begingroup$ No you're not missing anything. I think the point was for you to realize that retractions are "absolutely surjective" : any functor sends them to epimorphisms $\endgroup$ Commented Oct 15, 2019 at 9:57
  • $\begingroup$ @Max Ok, thank you very much! $\endgroup$ Commented Oct 15, 2019 at 11:02


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