Questions: Fundamental Group of the Circle

Question 1

A loop on S¹ winds 3 times counterclockwise, then 1 time clockwise. What is its winding number, and what does this tell you about its homotopy class?

• A. Winding number 4; it is in the same homotopy class as a loop that winds 4 times counterclockwise
• B. Winding number 2; it is homotopic to a loop that winds exactly twice counterclockwise
• C. Winding number 3; the clockwise winding doesn't subtract because winding numbers are always positive
• D. Winding number 0; forward and backward windings always cancel completely

Answer: (B)

The winding number is the *net* count: 3 counterclockwise minus 1 clockwise equals 2. Under the isomorphism π₁(S¹) ≅ ℤ, loops with the same net winding number are homotopic — they can be continuously deformed into each other. This loop is in homotopy class [2], the same class as a simple loop winding twice counterclockwise. The shape and path of the loop are irrelevant; only the net winding number determines the homotopy class.

Question 2

Why is π₁(S²) = 0 (the trivial group) while π₁(S¹) ≅ ℤ?

• A. S² is a 2-dimensional space and the fundamental group of an n-dimensional space is always trivial for n ≥ 2
• B. S² has no interior 'hole'; any loop drawn on the sphere's surface can slide over the top and shrink continuously to a point
• C. The fundamental group of S² equals ℤ² since it has two dimensions, but this simplifies to 0 by convention
• D. Loops on S² cannot be composed because the sphere is not a group under multiplication

Answer: (B)

S¹ has a hole — a loop that winds around the circle cannot be contracted to a point without leaving the space. S² has no such obstruction: any loop on the surface of a 2-sphere can be pulled toward one pole and shrunk to a point without tearing. This is what 'simply connected' means. Option A is wrong; ℝ² is 2-dimensional with trivial fundamental group, but the torus T² = S¹ × S¹ is 2-dimensional with π₁ = ℤ × ℤ. Dimension alone does not determine the fundamental group.

Question 3

A loop on S¹ that is geometrically complicated — winding forward and backward many times — is expected to have a nonzero winding number because its complexity prevents it from being contractible.

• T. True
• F. False

Answer: False

The winding number tracks *net* wrapping, not geometric complexity. A loop can wind 100 times clockwise and 100 times counterclockwise, producing an intricate path, yet have winding number 0 — placing it in the trivial homotopy class, homotopic to a constant loop. Only the net count determines the homotopy class. This is the key conceptual point: topology cares about what cannot be continuously undone, not about visual complexity.

Question 4

The composition of a loop with winding number 2 and a loop with winding number −3 represents an element of π₁(S¹) with winding number −1, consistent with the group operation corresponding to integer addition.

• T. True
• F. False

Answer: True

Under the isomorphism π₁(S¹) ≅ ℤ, loop composition corresponds precisely to integer addition. A loop of winding number 2 followed by a loop of winding number −3 gives net winding 2 + (−3) = −1. This is the content of the isomorphism: the algebraic structure of the fundamental group (loop composition) mirrors the arithmetic of the integers (addition), with winding number serving as the isomorphism.

Question 5

How does the covering space ℝ → S¹ (given by t ↦ e^{2πit}) make the winding number rigorous, and why does the integer endpoint of a lifted path capture the homotopy class of the loop?

Model answer: Given a loop γ in S¹ based at 1, the covering map lets us lift γ to a path γ̃ in ℝ starting at 0. Since the map t ↦ e^{2πit} identifies all integers with the basepoint 1, the lifted path must end at some integer n — the winding number of γ. Two loops are homotopic in S¹ if and only if their lifts end at the same integer, so the endpoint gives a well-defined homotopy invariant. Composition of loops corresponds to concatenation of lifts, and concatenated lifts add their endpoints — which is why composition in π₁(S¹) corresponds to addition in ℤ.

The covering space proof is what elevates the winding number from geometric intuition to a rigorous algebraic invariant. The key facts are: (1) lifts are unique given a starting point, (2) homotopic loops lift to paths with the same endpoint, and (3) the endpoint is always an integer. Together these establish the isomorphism π₁(S¹) ≅ ℤ rather than just an analogy.