Van Kampen's Theorem Statement

Algebraic Topology · Axiom Academy

LESSON Van Kampen's Theorem Statement Computing fundamental groups via decomposition 1. The Setup: Covering a Space Van Kampen's theorem begins with a topological space X that can be expressed as a union of two open, path-connected subspaces. Let U₁, U₂ be open, path-connected subsets of X Suppose U₁ ∩ U₂ is path-connected and non-empty Choose a basepoint x₀ ∈ U₁ ∩ U₂ The inclusion maps of the subspaces induce homomorphisms between their fundamental groups. ι₁: U₁ ↪ X induces (ι₁)₊: π₁(U₁, x₀) → π₁(X, x₀) ι₂: U₂ ↪ X induces (ι₂)₊: π₁(U₂, x₀) → π₁(X, x₀) j₁: U₁ ∩ U₂ ↪ U₁ induces (j₁)₊: π₁(U₁ ∩ U₂, x₀) → π₁(U₁, x₀) j₂: U₁ ∩ U₂ ↪ U₂ induces (j₂)₊: π₁(U₁ ∩ U₂, x₀) → π₁(U₂, x₀) Under the conditions above, the fundamental group of X is the amalgamated free product of the fundamental groups of U₁ and U₂. Statement: If X = U₁ ∪ U₂ where U₁, U₂ are open and path-connected, and U₁ ∩ U₂ is path-connected and non-empty, then: In other words, the fundamental group of X is determined by: The fundamental groups of the pieces (U₁ and U₂) How they overlap (their intersection U₁ ∩ U₂) Van Kampen's theorem has a beautiful geometric interpretation: loops in X can be broken down into loops in U₁ and U₂. Any loop γ in X based at x₀ can be subdivided into segments that lie entirely in U₁ or entirely in U₂. The transitions between these segments occur in the intersection U₁ ∩ U₂. Free Product Part: We can concatenate loops from U₁ and U₂

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