 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-2/21f/c/solution

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

Label the four vertices cyclically by

Z /4

. Reading the arrows in the diagram, an

a

-edge moves one step in one direction and a

b

-edge one step in the other. Thus the monodromy homomorphism is

ρ : F (a, b) ⟶ Z /4, ρ (a) = - 1, q u a d ρ (b) = 1

(up to simultaneously reversing the chosen numbering). By the Galois correspondence for covering spaces, the subgroup associated with the chosen vertex above

x_{0}

ker ρ = {w \in F (a, b) : #_{b} (w) - #_{a} (w) \equiv 0 (mod 4)} .

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