----- > [!proposition] Proposition. ([[fundamental groups and products commute]]) > Let $(X,x_{0})$ and $(Y,y_{0})$ be [[topological space|topological spaces]]. The [[product topology|product]] $X \times Y$ has [[fundamental group]] $\pi_{1}\big(X \times Y, (x_{0},y_{0}) \big)$ [[group isomorphism|isomorphic]] to the [[direct product of groups|direct product]] $\pi_{1}(X,x_{0}) \times \pi_{1}(Y,y_{0})$. > In particular, $\begin{align} \phi: \pi_{1}\big(X \times Y, (x_{0},y_{0})\big) & \to \pi_{1}(X,x_{0}) \times \pi_{1}(Y,y_{0}) \\ [\gamma] & \xmapsto{\phi} \big(p_{*}[\gamma], q_{*}(\gamma)\big) \end{align}$ is a [[well-defined]] [[group isomorphism]] with inverse $\begin{align} \psi: \pi_{1}(X,x_{0}) \times \pi_{1}(Y,y_{0}) & \to \pi_{1}\big(X \times Y, (x_{0},y_{0})\big) \\ ([\alpha], [\beta]) & \xmapsto{\psi} [\alpha \times \beta]. \end{align}$ Here, >- $p:X \times Y \to X$ is the [[projection function|projection map]] onto $X$, [[homomorphism of fundamental groups induced by a continuous map|inducing]] $p_{*}$; >- $q: X \times Y \to Y$ is the [[projection function|projection map]] onto $Y$, [[homomorphism of fundamental groups induced by a continuous map|inducing]] $q_{*}$; >- $\alpha \times \beta: [0,1] \to X \times Y$ is the [[parameterized curve|loop]] $(\alpha \times \beta)(t):=\big( \alpha(t), \beta(t) \big)$. ^ac2984 > [!proposition] Corollary. > The [[torus]] $\mathbb{S}^{1} \times \mathbb{S}^{1}$ has [[fundamental group]] [[group isomorphism|isomorphic]] to $\mathbb{Z}^{2}$, since [[the fundamental group of the circle is infinite cyclic]]. ^4e601d > [!proof]- Proof. ([[fundamental groups and products commute]]) > First we verify that $\phi$ is a [[group homomorphism]]:$\begin{align} \phi([\gamma_{1}] * [\gamma_{2}]) = & \phi([\gamma_{1} * \gamma_{2}]) \\ = & \big( p_{*}([\gamma_{1} * \gamma_{2}]), q_{*}([\gamma_{1} * \gamma_{2}]) \big) \\ = & \big( [p \circ (\gamma_{1} * \gamma_{2})], [q \circ (\gamma_{1} * \gamma_{2})] \big) \\ = & \big( [(p \circ \gamma_{1})] * [(p \circ \gamma_{2})] , [(q \circ \gamma_{1} )] * [(q \circ \gamma_{2})] \big) \\ = & \big( p_{*}([\gamma_{1}]) * p_{*}([\gamma_{2}]) , q_{*}([\gamma_{1}]) * q_{*}([\gamma_{2}])\big) \\ = & \big( p_{*}([\gamma_{1}]), q_{*}([\gamma_{1}]) \big) \ * \ \big( p_{*}([\gamma_{2}]), q_{*}(\gamma_{2}) \big) \\ = & \phi([\gamma_{1}]) * \phi([\gamma_{2}]); \end{align}$ note that we [[continuous functions respect path homotopy|employed this result]]. (There might be a more general notion of 'product homomorphism' at play here — we did not really use any topology to obtain this result, just group theory...) > Now we check that $\psi=\phi ^{-1}$. Write $\gamma:[0,1] \to X \times Y$ as $\gamma(t)=\big( \gamma_{X}(t), \gamma_{Y}(t) \big)$, so that $p \circ \gamma(t)=\gamma_{X}(t)$ and $q \circ \gamma(t)=\gamma_{Y}(t)$. Observe $(\gamma_{X} \times \gamma_{Y})(t)=\big( \gamma_{X}(t), \gamma_{Y}(t) \big)=\gamma(t)$. It follows that $\begin{align} \psi \circ \phi([\gamma])= & \psi\big( p_{*}[\gamma], q_{*}[\gamma] \big) \\ = & \psi\big( [p \circ \gamma], [q \circ \gamma] \big) \\ = & \psi([\gamma_{X}], [\gamma_{Y}]) \\ = & [\gamma_{X} \times \gamma_{Y}] \\ = & [\gamma]. \end{align}$ And $\begin{align} \phi \circ \psi([\alpha], [\beta])= & \phi([\alpha \times \beta]) \\ = & \big( p_{*}([\alpha \times \beta]), q_{*}([\alpha \times \beta]) \big) \\ = & \big( [\alpha], [\beta] \big). \end{align}$ This finishes the proof. ^878b7e ----- #### ---- #### 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 > ```