Fundamental Group of the Torus

Algebraic Topology · Axiom Academy

EXAMPLE Fundamental Group of the Torus Computing π₁(T²) ≅ ℤ × ℤ using the two generating loops and their abelian nature Excellent work! You've completed this example. Here's what we learned: Product space theorem: The fundamental group of a product space is the product of the fundamental groups: π₁(X × Y) ≅ π₁(X) × π₁(Y). Torus structure: The torus T² = S¹ × S¹ can be viewed as the product of two circles, giving us π₁(T²) ≅ π₁(S¹) × π₁(S¹) ≅ ℤ × ℤ. Two independent generators: The meridian loop (going around the "small circle") and longitude loop (going around the "big circle") generate all loops on the torus and are independent. Abelian nature: Unlike the figure-eight space (which has a free group), the torus has an abelian fundamental group because the two generators commute: a·b = b·a for loops a and b. General principle: Any loop on the torus can be expressed uniquely as going m times around the meridian and n times around the longitude, represented by the pair (m, n) ∈ ℤ × ℤ. This example demonstrates how product spaces have particularly nice fundamental groups!

This is the written version of the interactive lesson above. See the full Algebraic Topology course.