I am trying to understand better Open book decompositions.
To that effect, I tried to work out a couple of (relatively-simple) examples, specifically, for $\mathbb R^2 $ and higher.
But I have not been able to get much context to understand why this is important and when/where it matters.
Anyway, here is my work for $\mathbb R^2 $ (I am assuming a fibration to be just a continuous surjection):
We want a triplet ($\mathbb R^2, K, \pi $) , where $K$ is a submanifold of $\mathbb R^2$ of codimension-two (so dimension 0 here), and $\pi:(\mathbb R^2\backslash{(0,0)})\rightarrow S^1$ is a fibration.
Only map I can think of is $$\pi:(x,y) \rightarrow \frac{(x,y)}{\|(x,y)\|}.$$
Now, I think we can generalize, for $\mathbb R^n$ to $K$ as above being the $z$-axis,
(i.e., the subset $(0,0,z)$ ) and $$\pi:(x,y,z) \rightarrow \frac{(x,y,0)}{\|(x,y,0)\|}.$$
Is this correct? And could someone suggest some context so that I can understand why open book decompositions are important in general? I have found out some results like that of every closed manifold admitting an open book decomposition, and the existence of a bijection, for closed 3-manifolds, between contact structures and open book decompositions (up to isotopy, I think) , but would appreciate some more comments/refs.
Thanks for any help.