----- > [!proposition] Proposition. ([[inclusion map's pushforward morphism onto a retract is injective]]) > If $A$ is a [[retract]] of $X$, then the [[group homomorphism|homomorphism]] of [[fundamental group|fundamental groups]] [[homomorphism of fundamental groups induced by a continuous map|induced by]] [[inclusion map|inclusion]] is an [[injection]], $\pi_{1}(A,a) \xhookrightarrow{j_*} \pi_{1}(X,x_{0}).$ > [!proposition] Corollary. (No-Retraction Theorem) > There is no [[retract|retraction]] of $B^{2}$ onto $\mathbb{S}^{1}$. For if there were, an [[injection]] from $\pi_{1}(\mathbb{S}^{1},a) \cong \mathbb{Z}$ into $\pi_{1}(B^{2}, x_{0}) \cong (0)$ would exist. > \ > Alternatively, we can observe $r \circ \iota = \id_{\mathbb{S}^{1}}$, hence $r_{*} \circ \iota_{*}=\text{identity homomorphism}$, implying that there exists a factorization $\overbrace{\pi_{1}(\mathbb{S}^{1} ,1)}^{\cong \mathbb{Z}} \xrightarrow{\iota_{*}} \overbrace{\pi_{1}(\mathbb{D}^{2}, 1)}^{\cong (e)} \xrightarrow{r_{*}} \overbrace{\pi_{1}(\mathbb{S}^{1}, 1)}^{\cong \mathbb{Z}}$ > of the [[identity map|identity homomorphism]], which cannot happen. > [!proof]+ Proof. ([[inclusion map's pushforward morphism onto a retract is injective]]) > Suppose $r: X \to A$ is a [[retract|retraction]]. Then $r \circ j=\id_{A}$. So $r_{*} \circ j_{*}([f])=[e_{a}]$ for all $[f] \in \pi_{1}(A,a)$. So $j_{*}$ must be [[injection|injective]] because it has as [[inverse map|left-inverse]] $r_{*}$ (see [[characterization of injectivity and surjectivity in Set]].) ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = [](characterization%20of%20injectivity%20and%20surjectivity%20in%20Set.md)ataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch04