---- > [!definition] Definition. ([[fundamental group]]) > Let $X$ be a [[topological space]]; let $x_{0} \in X$. The set of [[path homotopy]] [[equivalence class|classes]] of [[parameterized curve|loops]] based at $x_{0}$, along with the [[fundamental groupoid|concatenation operation]] $*$, is called the **fundamental group** or **first homotopy group** of $X$ **relative to the base point $x_{0}$**. > \ > It is denoted by $\pi_{1}(X,x_{0})$. > [!note] Note. > The construction of the fundamental group is a [[covariant functor|functor]] from the [[category]] $\mathsf{Top}_{*}$ of [[pointed set|based]] [[topological space|topological spaces]] to $\mathsf{Grp}$. The morphism maps come from [[homomorphism of fundamental groups induced by a continuous map|here]]. ^note > [!note] Remark. > There is a notion of [[path-connected component|zeroth homotopy group]] $\pi_{0}(X)$ as well as [[higher homotopy group|higher homotopy groups]]. ^note > [!justification] > We must show that $\pi_{1}(X,x_{0})$ is indeed a [[group]]. This follows directly from the fact that the [[fundamental groupoid]] is a [[groupoid]]: once the base point $x_{0}$ is fixed, $*$ is well-defined for any two loops in $X$ about it, and therefore the [[partial function]] $*$ becomes a [[binary operation]]. All other group axioms follow as they did in [[fundamental groupoid]]. > [!basicexample] > The fundamental groups of $\mathbb{R}^{2} - \{ 0 \}$ and $\mathbb{S}^{1}$ are [[group isomorphism|isomorphic]]. It is clear that each is space is [[connected]] because both are [[path-connected]] ($\mathbb{R}^{2} - \{ 0 \}$ with 'broken lines' and $\mathbb{S}^{1}$ via identification with $[0,1)$). > Define $f:\mathbb{R}^{2} - \{ 0 \} \to \mathbb{S}^{1}$ to be the normalization map $f(v):=\frac{v}{\|v\|}$, and let $\iota:\mathbb{S}^{1} \to \mathbb{R}^{2} - \{ 0 \}$ be the [[inclusion map]]. > Note that $f \circ \iota$ is the [[identity map]] on $\mathbb{S}^{1}$, because $(f \circ \iota)(v)=f(v)=\frac{v}{\|v\|}=\frac{v}{1}=v$ for any $v$ on the [[unit circle]], since $\|v\|=1$. > Also note that $\iota \circ f$ and the [[identity map]] on $\mathbb{R}^{2}-\{ 0 \}$ are [[homotopy|homotopic]]. Indeed, the interpolation map $H(v,t)=(1-t) \frac{v}{\|v\|} + tv$ witnesses this (for the same reasoning as for the [[straight-line homotopy]]; this works because the lines will never pass through the origin). This does not seem to connect to the hint, so maybe it is wrong. > Finally we show that the two [[fundamental group]]s are [[group isomorphism|isomorphic]]. $f_{*}$ is a [[group homomorphism|homomorphism]] by [[homomorphism of fundamental groups induced by a continuous map]]. We claim that it has inverse $g_{*}:=\iota_{*}$. Indeed, let $\alpha$ be a [[parameterized curve|loop]] based at $x_{0}$. Since $f\circ \iota = \id$ we have $\begin{align} (f_{*} \circ g_{*})([\alpha]) = & (f \circ \iota)_{*}([\alpha])=i_{*}([\alpha])=[\alpha]. \end{align}$ And since $\iota \circ f$ is [[homotopy|homotopic]] to the [[identity map]] on $\mathbb{R}^{2} - \{ 0 \}$, as witnessed by some [[continuous]] using [[continuous functions respect path homotopy]] and that $\iota \circ f$ and $\id$ are both [[continuous]] we have $\begin{align} (\iota_{*} \circ f_{*})([\alpha])= & \iota_{*}([f \circ \alpha]) \\ = & [(\iota \circ f) \circ \alpha] \\ = & [\id \circ \alpha] \\ = & [\alpha]. \end{align}$ Thus the [[bijection|bijectivity]] of $f_{*}$ is shown. ^ca2f57 ---- #### ---- #### 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 > ```