Exercise 52.4

Question

Let $A \subset X$; suppose $r\colon X 	o A$ is a continuous map such that
$r(a) = a$ for each $a \in A$. (The map $r$ is called a
\emph{	extbf{retraction}} of $X$ onto $A$.) If $a_0 \in A$, show that
\begin{equation*}
  r_*\colon \pi_1(X,a_0) \longrightarrow \pi_1(A,a_0)
\end{equation*}
is surjective.

Amit Awasthi · Accepted Answer

\begin{proof}
  Letting $\iota\colon A \hookrightarrow X$ be the inclusion map, we see $r \circ \iota = \mathrm{id}_A$, and so by Theorem 52.4, $r_* \circ \iota_* = (r \circ \iota)_* = (\mathrm{id}_A)_* = \mathrm{id}_{\pi_1(A,a_0)}$. This implies $r_*$ is surjective.
\end{proof}

Exercise 52.4

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Exercise 52.4

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