---- > [!definition] Definition. ([[homomorphism of fundamental groups induced by a continuous map]]) > Let $X,Y$ be [[topological space|topological spaces]] and $h:X \to Y$ [[continuous]]. Fix $x_{0} \in X$ and write $h(x_{0})=:y_{0}$. Define $\begin{align} h_{*} & : \pi_{1}(X, x_{0}) \to \pi_{1}(Y,y_{0}) \\ h_{*}([f]) & := [h \circ f] . \end{align}$ $h_{*}$ is a [[group homomorphism|homomorphism]] of [[fundamental group|fundamental groups]] $\pi_{1}(X,x_{0})$ and $\pi_{1}(Y,y_{0})$, called the **homomorphism induced by $h$, relative to the base point $x_{0}$**. Sometimes written $\pi_{1}(h)$. > [!justification] > The map $h_{*}$ is [[well-defined]], for if $F$ is a [[path homotopy]] between the [[parameterized curve|paths]] $f$ and $f'$, then $h \circ F$ is a [[path homotopy]] between the [[parameterized curve|paths]] $h \circ f$ and $h \circ f'$. The fact that $h_{*}$ is a [[group homomorphism]] follows from the equation $(h \circ f) * (h \circ g)=h \circ (f * g).$ > See [[continuous functions respect path homotopy]] for more. > [!basicproperties] Functorial properties of the induced homomorphism. > 1. If $h:(X,x_{0}) \to (Y,y_{0})$ and $k:(Y,y_{0}) \to (Z,z_{0})$ are [[continuous]], then $(k \circ h)_{*}=k_{*} \circ h_{*}$; > 2. The [[identity map]] $i:(X,x_{0}) \to (X,x_{0})$ induces the identity homomorphism $i_{*}$. > [!proof] Proof of functorial properties. > >This is trivial. >1. Let $h,k$ be [[continuous]]. Then $\begin{align} (k \circ h)_{*}([f]) = & [(k \circ h) \circ f] \\ = & [k \circ (h \circ f)] \\ = & k([h \circ f]) \\ = & k_{*} ([h \circ f]) \\ = & k_{*}(h_{*}([f])) \\ = & (k_{*} \circ h_{*})([f]). \end{align}$ >2. Clearly $i_{*}([f])=[i \circ f]=[f]$. ---- #### ---- #### 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 = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > 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 > ```