[[Fundamental group]]
# Fundamental group preserves products

Let $(X_{1}, x_{1})$ and $(X_{2}, x_{2})$ be pointed spaces and $(X, x) =  (X_{1}, x_{1}) \times (X_{2}, x_{2})$ have the [[Product topology]] with the projections $p_{1} : X \twoheadrightarrow X_{1}$ and $p_{2} : X \twoheadrightarrow X_{2}$,
and let $\varpi_{1},\varpi_{2}$ denote the projections of the product group $\pi_{1}(X_{1},x_{1}) \times \pi_{2}(X_{2},x_{2})$.
Then there exists a unique isomorphism $\Phi$ such that the following diagram commutes:

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which is given by
$$
\begin{align*}
\Phi:
\pi_{1}(X,x) &\to \pi_{1}(X_{1},x_{1}) \times \pi_{1}(X_{2}, x_{2}) \\
[\alpha] &\mapsto ([p_{1} \alpha], [p_{2} \alpha]) = (\pi(p_{1})[\alpha], \pi(p_{2})[\alpha])
\end{align*}
$$

That is, the fundamental group of a [[Product topology]] is isomorphic to the [[Direct product of groups|direct product]] of fundamental groups. #m/thm/homotopy 

> [!check]- Proof
> From the [[Products and coproducts|universal property of the product]] $\Phi$ is a unique homomorphism.
> Let $\alpha$ be a a loop in $X$ with base $x$.
> If $\Phi[\alpha] = (e, e)$ then there exist homotopies $H_{1} : p_{1} \alpha \simeq c_{x_{1}}$ and $H_{2} : p_{2} \alpha \simeq c_{x_{2}}$.
> Thus $\alpha \simeq c_{x}$ by the homotopy
> $$
> \begin{align*}
> H(s,t) = (H_{1}(s,t), H_{2}(s,t))
> \end{align*}
> $$
> and hence $[\alpha] = e$,
> hence $\ker \Phi = \{e\}$ and [[Group monomorphism|thus]] $\Phi$ is injective.
> Now let $\alpha_{i}$ be a loop in $X_{i}$ with base $x_{i}$ for $i=1,2$.
> Then the following is a loop in $X$ with base $x$
> $$
> \begin{align*}
> \alpha : s \mapsto (\alpha_{1} (s), \alpha_{2}(s))
> \end{align*}
> $$
> and $\Phi[\alpha] = ([p_{1} \alpha], [p_{2}\alpha]) = ([\alpha_{1}], [\alpha_{2}])$.
> Hence $\Phi$ is surjective.
> <span class="QED"/>

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