Bouquet of Circles

Algebraic Topology · Axiom Academy

The fundamental group of n circles joined at a point is the free group F_n A bouquet of n circles , denoted ⋁ⁿ S¹, is formed by taking n copies of the circle S¹ and identifying a single point from each circle to form a common basepoint. Each circle contributes one "petal" to the bouquet, and all petals share the central basepoint. The fundamental group of a bouquet of n circles is the free group on n generators , denoted F_n. Each circle contributes exactly one generator to the free group. These generators have no relations between them, meaning any word formed from these generators represents a distinct element of the group. π₁(S¹) = F₁ ≅ Z (integers under addition) n = 2: Figure-eight (two circles) π₁(S¹ ∨ S¹) = F₂ = ⟨a, b⟩ (words like aba⁻¹b², abab⁻¹, etc.) π₁(⋁³ S¹) = F₃ = ⟨a, b, c⟩ (words like abc, a²bc⁻¹, etc.)

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