Questions: Homotopy of Paths

Question 1

Let X = ℝ² minus the origin (the punctured plane). Path γ loops once around the origin; path σ goes from the same start to the same end without encircling the origin. Which statement is correct?

• A. γ and σ are homotopic because they share the same endpoints
• B. γ and σ are homotopic because both are continuous functions on [0,1]
• C. γ and σ are not homotopic because any continuous deformation would have to pass through the missing origin
• D. γ and σ are not homotopic because their domains are different

Answer: (C)

Homotopy requires a *continuous* map H: [0,1]×[0,1] → X — but X is the plane with the origin removed. Any attempt to continuously deform γ into σ would force some intermediate path to cross the origin, which is not in X. Shared endpoints (option A) are necessary but not sufficient for homotopy. The topological obstruction — the hole — is exactly what homotopy theory is designed to detect.

Question 2

In a homotopy H: [0,1]×[0,1] → X between paths γ and σ from p to q, which conditions must hold?

• A. H(s,0) = γ(s) and H(s,1) = σ(s), and H(0,t) = p and H(1,t) = q for all s,t
• B. H(0,t) = γ(t) and H(1,t) = σ(t), and H(s,0) = p and H(s,1) = q for all s,t
• C. H(s,0) = p and H(s,1) = q for all s, with no constraint on H(0,t) or H(1,t)
• D. H must be a bijection from [0,1]×[0,1] onto X

Answer: (A)

The first coordinate s is position along the path; the second coordinate t is deformation time. At t=0 you traverse γ, at t=1 you traverse σ. The endpoint conditions H(0,t) = p and H(1,t) = q for all t ensure that throughout the entire deformation the starting and ending points remain fixed — the paths are not allowed to slide their endpoints. Option B confuses the roles of the two coordinates.

Question 3

In the full plane ℝ², any two paths with the same endpoints are homotopic.

• T. True
• F. False

Answer: True

True. ℝ² is convex: for any two paths γ and σ with the same endpoints, the straight-line interpolation H(s,t) = (1−t)γ(s) + tσ(s) is a valid homotopy. Every point along every intermediate path stays in ℝ², and the endpoints are clearly fixed throughout. The existence of holes or other topological obstructions is what can prevent homotopy; without them, all paths between the same endpoints are homotopic.

Question 4

A homotopy H: [0,1]×[0,1] → X is expected to satisfy H(s,0) = H(s,1) for most s.

• T. True
• F. False

Answer: False

False. H(s,0) = γ(s) is the initial path and H(s,1) = σ(s) is the final path — these are *different* paths (that's the whole point of the homotopy). The conditions that must hold are the *endpoint conditions*: H(0,t) = γ(0) = σ(0) and H(1,t) = γ(1) = σ(1) for all t, keeping the starting and ending points fixed throughout the deformation. Confusing path-time and deformation-time conditions is a common source of error.

Question 5

Why is the product topology on [0,1]×[0,1] the right structure to use in the definition of homotopy, rather than treating the two coordinates independently?

Model answer: A homotopy is a single continuous function H defined on the entire square [0,1]×[0,1]. Continuity of H means that small changes in *both* coordinates (s, t) simultaneously produce small changes in H(s,t) — the deformation is continuous not only in path position and in time separately, but jointly. The product topology on [0,1]×[0,1] is exactly the topology that makes this joint continuity precise, since open sets in the product are generated by products of open sets in each factor. Treating the coordinates independently would only guarantee continuity in each variable separately (continuity in s for fixed t, and in t for fixed s), which is a weaker condition that does not capture the geometric idea of a smooth, simultaneous deformation.

This distinction matters because joint continuity is required for the deformation to be truly smooth in both directions at once. A function that is continuous in each variable separately but not jointly could have pathological behavior — for instance, it could 'jump' as you move diagonally across the square. The product topology ensures the full geometric notion of a continuously varying family of paths.