I have the following questions:
Let $p:\Bbb{T}^2\to \Bbb{S}^1$ be a circle bundle. Is $p$ trivial ?
Before I show you what I attempted, I just want to let you know that I don't have any knowledge of fiber bundles. I just base my intuition on the little I know about vector bundles, and from what I quiclky read on Wikipedia I think that I'm allowed to do what follows.
Take the quotient map $q:[0,1]\to S^1$. We can pull back the bundle $p$ via $q$. Because $[0,1]$ is contractible, this pulled back bundle is trivial so we have $$\begin{array} {ccc} [0,1]\times \Bbb{S}^1 & \stackrel{h}{\longrightarrow} & E=\Bbb{T}^2\\ p' \Big\downarrow & & \Big\downarrow p\\ [0,1] & \stackrel q {\longrightarrow} & S^1 \end{array}$$
Then I guess I should glue the sides of the cylinder $[0,1]\times \Bbb{S}^1$ by factorizing $h$ but I don't know how to do it formally (I don't see how $f$ factors). Also should I have taken the universal cover $\mathbb{R}\to \Bbb{S}^1$ instead of $q$?
Second question: I am looking for a (short) introductory book to learn about the theory of fiber bundles. I don't know much about it so I can't really be specific in my request, but I'd like something which is "differentiable manifolds related". From what I have read I think that I'm looking for something like the 3rd chapter of Dale Husemoller, Fiber Bundles. I also know about Norman Steenford The topology of fibre bundles, but I'm not sure if this one is exactly what I'm looking for. Do you have any advices? (I've had a course of differential topology already but we didn't study fiber bundles)
Thanks in advance!