Quotient of a cylinder by the product of a rotation and translation

Question

Let $D$ be the unit disk, and $$f : D \times \mathbb R \rightarrow D \times \mathbb R$$ be defined by $$f(z,t)=(e^{2i\pi \alpha}z, t+1)$$ where $\alpha$ is an irrational number. Now consider $X = (D \times \mathbb{R})/f$.

Since $f$ acts discretely, $X$ is $3$-manifold. Also it somehow fibers above the circle. Is there a description of $X$ ?

Answer 1

Yes, it's a filled-in torus $D\times S^1$. The map $$\phi(z,t) = (e^{2\pi i \alpha t}z,t)$$ is a homeomorphism of $D\times \mathbb{R}$ onto itself, which conjugates $f$ to $g(z,t) = (z,t+1)$: $$ \phi^{-1}(f(\phi(z,t)) = (z,t+1) $$
The quotient by $g$ is $D\times S^1$.