 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-2/21f/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

*

, loops in

U (2)

(α \cdot β) (t) = α (t) β (t) .

Continuity of matrix multiplication makes this a loop, and pointwise multiplication of homotopies shows that it is well-defined on homotopy classes. Both operations have the constant loop

e (t) = I

as identity, and they satisfy the interchange law

(α * β) \cdot (γ * δ) = (α \cdot γ) * (β \cdot δ) .

For homotopy classes

a, b

, the interchange law gives

a \cdot b = (a * e) \cdot (e * b) = (a \cdot e) * (e \cdot b) = a * b .

It then gives

a * b = a \cdot b = (e * a) \cdot (b * e) = (e \cdot b) * (a \cdot e) = b * a .

Thus loop concatenation is commutative on homotopy classes, and

π_{1} (U (2), I) is abelian .

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