Fundamental Group π₁

Loop spaces, homotopy classes, and the fundamental group of topological spaces

Space + Loops

Homotopy parameter t 0.00

Fundamental Group

Loop Composition

γ·δ = first traverse γ, then δ

Winding Numbers

π₁(X,x₀) = {loops at x₀ up to homotopy} with concatenation — van Kampen's theorem

The fundamental group π₁(X) captures holes by classifying loops (based at a point x₀) up to continuous deformation (homotopy). π₁(circle) = ℤ (integer winding number), π₁(ℝ²) = 0 (contractible), π₁(ℝ²∖{0}) = ℤ (one hole), π₁(two holes) = F₂ (free group on 2 generators — non-abelian!), π₁(torus) = ℤ×ℤ. Van Kampen's theorem: π₁(X∪Y) = π₁(X) *_{π₁(X∩Y)} π₁(Y) (amalgamated free product). The homotopy slider shows how a loop can be continuously deformed.