Find a group \(G\) of homeomorphisms of the torus \(T\) having order 2 such that \(T/G\) is homeomorphic to the torus.
Find a group \(G\) of homeomorphisms of \(T\) having order 2 such that \(T/G\) is homeomorphic to the Klein bottle.
Solution
Write the torus as \(T=\mathbb R^2/\mathbb Z^2\), with coordinates \([x,y]\).
Let \[ \tau[x,y]=[x+1/2,y]. \] Then \(\tau^2=1\) and \(\tau\) has no fixed points. The quotient identifies the first coordinate modulo \(1/2\), so \[ T/\langle\tau\rangle\cong (\mathbb R/(\tfrac12\mathbb Z))\times(\mathbb R/\mathbb Z)\cong T. \] Thus \(G=\{1,\tau\}\) has order \(2\) and torus quotient.
Let \[ \sigma[x,y]=[x+1/2,-y]. \] Again \(\sigma^2=1\), and it is fixed-point-free because a fixed point would require \(x+1/2\equiv x\pmod1\). A fundamental domain is the strip \(0\le x\le1/2\). Its vertical sides are identified by \[ (0,y)\sim(1/2,-y), \] while \(y\) is periodic modulo \(1\). This is precisely the standard square model of the Klein bottle. Hence \[ T/\langle\sigma\rangle\cong K. \]