---- > [!definition] Definition. ([[homotopy relative to a subset]]) > Let $X,Y$ be [[topological space|topological spaces]] and $f,g:X \to Y$ two [[continuous]] maps. $H: X\times I \to Y$ is called a **homotopy relative to a subset $A \subset X$** if it is a [[homotopy]] and if $f(a)=g(a)=H(a,t)$ for all $a \in A$ and $t \in I$. (I.e., $H$ keeps the subspace fixed.) > \ > A **based homotopy** is a [[homotopy]] relative to some base point $x_{0}$. > [!basicexample] > Let $f,g:[0,1]\to X$ be [[parameterized curve|paths]] in $X$ from $x_{0}$ to $x_{1}$. Any [[path homotopy]] between $f$ and $g$ is automatically a [[homotopy]] relative to the subset $\{ 0,1 \} \subset [0,1]$. Let $X,Y$ be [[topological space|topological spaces]], $x_{0}\in X$ be arbitrary, and $f,g:X \to Y$ be two [[homotopy|homotopic]] (witnessed by $H$) [[continuous]] maps. Let $\alpha_{0}(t)=H(x_{0}, t)$, which defines a [[parameterized curve]] $\alpha_{0}:I \to Y$ from $H(x_{0},0)=f(x_{0})$ to $H(x_{0},1)=g(x_{0})$. Then $f_{*}([\alpha])=[\alpha_{0}]* g_{*}([\alpha])*[\overline{\alpha_{0}}]$ for any [[parameterized curve|loop]] $\alpha$ in $X$ based at $x_{0}$. (Here, $f_{*}$ denotes the [[homomorphism of fundamental groups induced by a continuous map|homomorphism induced by the continuous map]] $f$, and $*$ is the [[fundamental groupoid|path concatenation operation]].) ---- #### ---- #### 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 > ```