Give the definitions of covering space and deck transformation (or covering transformation).
Describe the universal cover of the Klein bottle and its group of deck transformations.
Explicitly give a collection of deck transformations on \(\{(x, y) \mid -1 \leq x \leq 1, -\infty < y < \infty\}\) such that the quotient is a Möbius band.
Find the universal cover of \(\mathbb{RP}^2 \times S^1\) and explicitly describe its group of deck transformations.
Solution
1. A covering map is a surjective continuous map \(p:\widetilde X\to X\) such that every \(x\in X\) has an open neighborhood \(U\) for which\[ p^{-1}(U)=\bigsqcup_{\alpha}U_\alpha \]and every restriction \(p|_{U_\alpha}:U_\alpha\to U\) is a homeomorphism. A deck transformation is a homeomorphism \(h:\widetilde X\to\widetilde X\) satisfying \(p\circ h=p\).
2. For the Klein bottle, take \(\widetilde K=\mathbb R^2\) and let\[ T(x,y)=(x,y+1),\qquad R(x,y)=(x+1,-y). \]
2.1. The group \(\Gamma=\langle R,T\rangle\) acts freely and properly discontinuously on \(\mathbb R^2\), and a fundamental rectangle has the standard Klein-bottle edge identifications.
2.2. Moreover\[ RTR^{-1}=T^{-1}, \]so\[ \Gamma\cong\langle R,T\mid RTR^{-1}=T^{-1}\rangle\cong\pi_1(K). \]Hence \(\mathbb R^2\to\mathbb R^2/\Gamma\cong K\) is the universal covering and \(\Gamma\) is its deck group.
3. On \([-1,1]\times\mathbb R\), define\[ \tau(x,y)=(-x,y+1). \]Then\[ \tau^m(x,y)=((-1)^m x,y+m), \]and \(\langle\tau\rangle\cong\mathbb Z\) acts freely and properly discontinuously. A fundamental domain is \([-1,1]\times[0,1]\), with\[ (x,0)\sim(-x,1), \]so the quotient is a Möbius band.
4. The universal cover of \(\mathbb{RP}^2\times S^1\) is\[ S^2\times\mathbb R\longrightarrow\mathbb{RP}^2\times S^1, \qquad (v,t)\longmapsto([v],e^{2\pi i t}). \]Its deck transformations are\[ (v,t)\longmapsto((-1)^\varepsilon v,t+m), \qquad (\varepsilon,m)\in\mathbb Z/2\times\mathbb Z. \]Therefore\[ \operatorname{Deck}(S^2\times\mathbb R/\mathbb{RP}^2\times S^1) \cong\mathbb Z/2\times\mathbb Z. \]