Closed Curves

Closed Curves on Surfaces

The fundamental group π₁ classifies closed curves up to continuous deformation. On the torus T², π₁(T²) ≅ ℤ × ℤ — each curve is labeled by two winding numbers (p, q).

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Longitude

Wraps once around the large circle (toroidal direction). This is the other generator of π₁(T²).

Fundamental Group

π₁(T²) ≅ ℤ × ℤ

Generators: a (longitude), b (meridian)

Relation: aba⁻¹b⁻¹ = 1 (abelian!)

Current: (1, 0)