Figure Eight Space

Algebraic Topology · Axiom Academy

Computing the fundamental group using Van Kampen's Theorem Excellent work! You've computed the fundamental group of the figure-8 space. Here's what we learned: Decomposition Strategy: The figure-8 can be decomposed into two overlapping open sets, each containing one circle, with a contractible intersection. Van Kampen's Theorem: When the intersection is path-connected, π₁(X) is the free product of the fundamental groups of the open sets, amalgamated over the intersection. Free Product: Since the intersection is contractible (π₁(U ∩ V) is trivial), we get π₁(X) = ℤ * ℤ, the free group on two generators. Geometric Intuition: Each loop in the figure-8 can be decomposed into words in the two generators (loops around each circle), with no relations imposed. This demonstrates the power of Van Kampen's Theorem in computing fundamental groups by breaking complex spaces into simpler pieces!

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