In differential geometry we reached the topic of cocycles and sections, but we only covered the definitions. Due to this I am trying to solve some problems of a given study problem list, so I can fully understand this topic. The problem that I got stuck with is the following:
"Find the cocycle for the covering $\pi:S^{1}\rightarrow S^{1}$ such that $\pi(z)=z^{2}$ with $F=\mathbb{Z}_{2}=\mathbb{Z}/2\mathbb{Z}$. Does this covering have sections ?"
To be honest, I dont know where to begin regarding the cocycle, as this is the first time seeing such definition and there were examples given in class. On the other hand my intuition says to me that this covering has no sections, but as well I am not sure on how to proceed. Any kind of help would be really appreciated.