----- > [!proposition] Proposition. ([[fundamental group of wedge of n circles]]) > Let $X$ be the [[wedge sum|wedge]] of circles $S_{1},\dots,S_{n}$ with common point $p$. Then $\pi_{1}(X,p)$ is a [[free group]]. If $f_{i}$ is a [[parameterized curve|loop]] in $S_{i}$ that represents a [[generating set of a group|generator]] of $\pi_{1}(S_{i},p)$, then the [[parameterized curve|loops]] $f_{1},\dots,f_{n}$ represent a system of free generators for $\pi_{1}(X,p)$. > [!proof]- Proof. ([[fundamental group of wedge of n circles]]) > We induct on $n$. The result is immediate for $n=1$. Suppose it holds for $n$. Let $X$ be the wedge of circles $S_{1},\dots,S_{n}$ with $p$ the common point of these circles. Choose a point $q_{i}$ of $S_{i}$ different from $p$, for each $i$. Set $W_{i}:=S_{i}-q_{i}$, and let $\begin{align} > U= & S_{1} \cup W_{2} \cup \dots \cup W_{n} \text{ and } V=W_{1} \cup S_{2} \cup \dots \cup S_{n}. > \end{align}$ > Then $U \cap V=W_{1} \cap \dots \cap W_{n}$ (see Munkres' figure 71.1). Each of $U,V,U\cap V$ is [[path-connected]], by [[arbitrary union of nontrivially-intersecting connected subspaces is connected|this result applied to path-connectedness]]. > > Each $W_{i}$ is [[homeomorphism]] to an [[open interval]] and so thus has $p$ as a [[deformation retract]], let $F_{i}: W_{i} \times I \to W_{i}$ be a [[deformation retract|deformation retraction]] witnessing this. The maps $F_{i}$ fit together to form a [[deformation retract|deformation retraction]] of $U \cap V$ onto $p$. It follows that $U \cap V$ is [[simply connected]]. The result then follows from Corollary 70.3. ----- #### ---- #### 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 > ``` #reformatrevisebatch04