Seifert-Van Kampen-Brown theorem

Let $𝑋$ be a topological space with open cover ${𝑈, 𝑉}$ . Then the following is a fibre coproduct in $𝖳 𝗈 𝗉$ on the left and in $𝖦 𝗋 𝗉 𝖽$ on the right.¹² homotopy

where $𝑖 1, 𝑖 2, 𝑗 1, 𝑗 2$ denote natural inclusions; i.e. the Fundamental groupoid of $𝑋$ is a fibre coproduct of the fundamental groupoids of the open covering spaces $𝑈$ and $𝑉$ .

Proof

For the left diagram see Fibre products and coproducts in Top. Now suppose $(𝐺, 𝑓 1, 𝑓 2)$ fits into the following diagram.

We must show the existence of a unique $𝑓$ such that the diagram commutes.

Uniqueness is the easier part to prove: For objects (points), if $𝑥 \in 𝑈$ then $𝑓 𝑥 = 𝑓 1 𝑥$ ; if $𝑥 \in 𝑉$ then $𝑓 𝑥 = 𝑓 2 𝑥$ ; and if $𝑥 \in 𝑈 \cap 𝑉$ the assignments agree. For a homotopy path $[𝛾] \in 𝜋 1 𝑋 (𝑥, 𝑦)$ uniqueness follows from a representative $𝛾 : 𝕀 \to 𝑋$ . Using a Lebesgue number, $𝕀$ may be evenly subdivided into sections either entirely in either $𝛾 - 1 𝑈$ or $𝛾 - 1 𝑉$ , giving paths $𝛾 1, \dots, 𝛾 𝑛$ where $𝛾 ≃ 𝛾 1 \dots 𝛾 𝑛$ Thus $𝑓 [𝛾]$ must agree with applying $𝑓 1$ and $𝑓 2$ to each component path, which is clearly invariant under refinement and therefore independent of the precise decomposition.

For existence, we need to show that $𝑓$ is independent of the representative $𝛾$ . Let $𝛾 0 ≃ 𝛾 1$ by virtue of a homotopy of paths $𝐻 : (𝑡, 𝑠) \mapsto 𝛾 𝑠 (𝑡)$ . Once again a Lebesgue number may be used to divide $𝕀 2$ into a $𝑘 \times 𝑘$ grid such that each box is entirely in either $Γ - 1 𝑈$ or $Γ - 1 𝑉$ . Assign to the box with bottom-left corner at $(𝑖 𝑘, 𝑗 𝑘)$ the paths $𝑎 𝑗, 𝑖, 𝑎 𝑗 + 1, 𝑖 : 𝕀 \to 𝕀 2$ rightwards along its top and bottom edges respectively, and $𝑐 𝑖, 𝑗, 𝑐 𝑖, 𝑗 + 1 : 𝕀 \to 𝕀 2$ upwards along its left and right edges respectively. Clearly $𝑐 𝑖, 𝑗 \cdot 𝑎 𝑗 + 1, 𝑖 ≃ 𝑎 𝑗, 𝑖 \cdot 𝑐 𝑖 + 1, 𝑗$ as paths, and $𝑎 𝑗, 𝑖 ≃ 𝑐 𝑖, 𝑗 \cdot 𝑎 𝑗 + 1, 𝑖 \cdot ―――― 𝑐 𝑖 + 1, 𝑗$ . Since $Γ 𝑐 0, 𝑗 / 𝑘$ and $Γ 𝑐 1, 𝑗 / 𝑘$ are constant paths in either $𝑈$ or $𝑉$ , applying $Γ$ to get paths in $𝑋$ for each $𝑗 = 0, \dots, 𝑘$
$𝑓 𝛾 𝑗 / 𝑘 = 𝑘 ⨀ 𝑖 = 0 𝑓 ◻ Γ 𝑎 𝑗 / 𝑘, 𝑖 / 𝑘 = 𝑘 ⨀ 𝑖 = 0 𝑓 ◻ Γ 𝑐 𝑖 / 𝑘, 𝑗 / 𝑘 ⊙ 𝑓 ◻ Γ 𝑎 (𝑗 + 1) / 𝑘, 𝑖 / 𝑘 ⊙ 𝑓 ◻ Γ ――――― 𝑐 (𝑖 + 1) / 𝑘, 𝑗 / 𝑘 = 𝑓 ◻ Γ 𝑐 0, 𝑗 / 𝑘 ⊙ (𝑘 ⨀ 𝑖 = 0 𝑓 ◻ Γ 𝑎 (𝑗 + 1) / 𝑘, 𝑖 / 𝑘) ⊙ 𝑓 ◻ Γ ――― 𝑐 1, 𝑗 / 𝑘 = 𝑘 ⨀ 𝑖 = 0 𝑓 ◻ Γ 𝑎 (𝑗 + 1) / 𝑘, 𝑖 / 𝑘 = 𝑓 𝛾 (𝑗 + 1) / 𝑘$
where $𝑓 ◻$ denotes applying $𝑓 1$ or $𝑓 2$ depending on whether a path is in $𝑈$ or $𝑉$ . It follows from $𝑘$ iterations that $𝑓 𝛾 0 = 𝑓 𝛾 1$ .

The classical Seifert-Van Kampen theorem concerns the Fundamental group, which can easily be derived from the above theorem. Ronald Brown introduced the groupoid formulation.

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2020, Topology: A categorical approach, §6.7, pp. 139–140 ↩
2006, Topology and groupoids, §6.7, pp. 240ff ↩

Seifert-Van Kampen-Brown theorem

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Seifert-Van Kampen-Brown theorem

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