Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 11G / d

Solution

A loop whose Klein deck transformation has odd

c

has one closed lift to the orientation double cover; one with even

c

has two. Thus the preimage of

γ_{1}

contains one closed geodesic, while those of

γ_{2}

and

γ_{3}

contain two each.

In the torus unit square, representatives are

γ_{1} (t) = [t, 0],

which maps twice around

γ_{1}

;

γ_{2, 1} (t) = [0, t], γ_{2, 2} (t) = [1/2, t];

and

γ_{3, 1} (t) = [t, t], γ_{3, 2} (t) = [t, 1/2 - t] .

All coordinates are taken modulo one. Hence the requested numbers are

1, 2, 2 .

The two diagonal lifts of

γ_{3}

meet over its self-intersection point but remain distinct closed geodesics.

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