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

If

(x, y) \sim (x + a, y + b)

in the torus, then

π (x + a, y + b) = (2 x + 2 a, y + b) .

This is Klein-equivalent to

(2 x, y)

by taking

(c, d) = (2 a, b)

. Hence

π

descends to a well-defined continuous map

π : T^{2} \to K, π ([x, y]) = [2 x, y] .

The deck group of the Klein bottle has an index-two subgroup consisting of transformations with even

c

; it is generated by translations

(x, y) \mapsto (x + 2, y)

and

(x, y) \mapsto (x, y + 1)

. This is exactly the image under

π

of the torus deck lattice. Therefore the orientation double cover of the Klein bottle has two sheets:

π : T^{2} \to K is 2 : 1 .

Solved by gpt-5.6-sol high.

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