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 $𝑝 1 : 𝑋 ↠ 𝑋 1$ and $𝑝 2 : 𝑋 ↠ 𝑋 2$ , and let $𝜛 1, 𝜛 2$ denote the projections of the product group $𝜋 1 (𝑋 1, 𝑥 1) \times 𝜋 2 (𝑋 2, 𝑥 2)$ . Then there exists a unique isomorphism $Φ$ such that the following diagram commutes:

which is given by

Φ : 𝜋 1 (𝑋, 𝑥) \to 𝜋 1 (𝑋 1, 𝑥 1) \times 𝜋 1 (𝑋 2, 𝑥 2) [𝛼] \mapsto ([𝑝 1 𝛼], [𝑝 2 𝛼]) = (𝜋 (𝑝 1) [𝛼], 𝜋 (𝑝 2) [𝛼])

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

Proof

From the universal property of the product $Φ$ is a unique homomorphism. Let $𝛼$ be a a loop in $𝑋$ with base $𝑥$ . If $Φ [𝛼] = (𝑒, 𝑒)$ then there exist homotopies $𝐻 1 : 𝑝 1 𝛼 ≃ 𝑐 𝑥 1$ and $𝐻 2 : 𝑝 2 𝛼 ≃ 𝑐 𝑥 2$ . Thus $𝛼 ≃ 𝑐 𝑥$ by the homotopy

𝐻 (𝑠, 𝑡) = (𝐻 1 (𝑠, 𝑡), 𝐻 2 (𝑠, 𝑡))

and hence $[𝛼] = 𝑒$ , hence $k e r Φ = {𝑒}$ and thus $Φ$ is injective. Now let $𝛼 𝑖$ be a loop in $𝑋 𝑖$ with base $𝑥 𝑖$ for $𝑖 = 1, 2$ . Then the following is a loop in $𝑋$ with base $𝑥$

𝛼 : 𝑠 \mapsto (𝛼 1 (𝑠), 𝛼 2 (𝑠))

and $Φ [𝛼] = ([𝑝 1 𝛼], [𝑝 2 𝛼]) = ([𝛼 1], [𝛼 2])$ . Hence $Φ$ is surjective.

#state/tidy | #lang/en | #SemBr