Fundamental Group of the Torus

Topology · Axiom Academy

EXAMPLE Fundamental Group of the Torus Computing π₁(T²) = ℤ × ℤ through the product space S¹ × S¹ Compute the fundamental group of the torus . We'll use the fact that the torus can be expressed as a product of two circles: , and apply theorems about fundamental groups of product spaces. You've successfully computed the fundamental group of the torus using the product space structure. The torus T² = S¹ × S¹ is topologically a product of two circles The fundamental group of S¹ is ℤ, representing winding numbers For product spaces: π₁(X × Y) ≅ π₁(X) × π₁(Y) (under mild conditions) Therefore π₁(T²) ≅ ℤ × ℤ, with two independent generators The generators correspond to loops going around the "meridian" and "longitude" of the torus Unlike the fundamental group of the sphere (which is trivial), the torus has a rich fundamental group This result generalizes: the n-torus has fundamental group ℤⁿ

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