I have a very simple question:
What is the mathematical definition of the spin group $\operatorname{Spin}(2)$?
According to Wikipedia it fits into a short exact sequence $$1\to \mathbb{Z}_2\to \operatorname{Spin}(2)\to S^1\to 1.$$ Thus, as opposed to $\operatorname{Spin}(n)$ for $n>2$, it is not the universal covering group of $\operatorname{SO}(2)=S^1$. One possibility is to take $\operatorname{Spin}(2)=S^1$ with the map $$\operatorname{Spin}(2)\to S^1,\quad t\mapsto t^2.$$ But in general group extensions are not unique so I am not sure if this is the definition of $\operatorname{Spin}(2)$.
A compact connected $1$-manifold is $S^1$, as $Spin(2)$ is the double cover of $S^1$ it is a compact and connected 1-manifold, so it is $S^1$.