 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2011/ii/paper-4/2f/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Put

z (t) = e^{2 πi t}

and use the straight homotopy

H (s, t) = z (t) + s (γ (t) - z (t))

. For

0 \leq s < 1

∣ H (s, t) ∣ \geq 1 - s ∣ γ (t) - z (t) ∣ \geq 1 - s > 0.

s = 1

it is nonzero by the hypothesis. Thus this is a homotopy of closed loops avoiding the origin, even though the given distance bound is non-strict. The homotopy invariance of winding number gives

w (γ; 0) = w (e^{2 πi t}; 0) = 1.

 New to topics? Read the docs here!