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, thenThis is Klein-equivalent to $(2x,y)$ by taking $(c,d)=(2a,b)$. Hence $\tilde{\pi }$ descends to a well-defined continuous map

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 $\tilde{\pi }$ of the torus deck lattice. Therefore the orientation double cover of the Klein bottle has two sheets: