I am having a hard time proving this although it looks trivial…
Let $r:X\to A$ be a retraction between a topological space $X$ and $A\subset X$ such that $r(a_0)=a_0$ for $a_0\in A$
then the induced homomorphism $r_*:\pi_1(X,a_0)\to \pi_1(A,a_0)$ is surjective.
I tried to prove it as follows:
I showed that if $g$ is a loop in $A$ based at $a_0$ then it is a loop also in $X$ based at $a_0$
So, given $[g]\in \pi_1(A,a_0)$, let us take $[g]\in \pi_1(X,a_0)$ (which will stand for a different homotopy class)
So we get: $$r_*([g])=[r\circ g]=[id_A\circ g]=[g]$$ that's since $Im(g)\subset A$ and $r$ is a retraction.
I believe that something is off in that proof in the part with the homotopy classes, so help please
Best Answer
Consider $i$ canonical injection $i\colon A \to X$. The fact that $r$ is a retract means $$r \circ i = \mathrm{id}_A$$ From this we get $$r_* \circ i_* = \mathrm{id}_{\pi_1(A)}$$ and this implies $r_*$ surjective since it has a right inverse.