Wedge Sums

Algebraic Topology · Axiom Academy

Joining topological spaces at a single point and computing fundamental groups Given two pointed spaces (X, x₀) and (Y, y₀), the wedge sum X ∨ Y is the quotient space obtained by taking the disjoint union of X and Y and identifying the basepoints x₀ ~ y₀. 2. Fundamental Group of Wedge Sum When the intersection is simply connected (just a point), the fundamental group of the wedge sum is the free product of the fundamental groups. Example 1: S¹ ∨ S¹ - Two circles joined at a point (figure-eight) π₁(S¹ ∨ S¹) = Z * Z = F₂ (free group on two generators) Example 2: S² ∨ S¹ - A sphere and circle joined at a point π₁(S² ∨ S¹) = π₁(S²) * π₁(S¹) = e * Z ≅ Z (since S² is simply connected)

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