Double cover of $SO(2)$

Question

In most textbooks i found that $SPIN(2)$ group is isomorphic to $U(1)$. That is $U(1)$ is double cover of $SO(2)$. But $U(1)$ is isomorphic to $SO(2)$. So how can it be? Does this definition of double cover does not work for $n=2$ ?

Answer 1

What's the problem? You can see $U(1)$ as a double cover of $SO(2,\mathbb{R})$:$$\begin{array}{ccc}U(1)&\longrightarrow&SO(2,\mathbb{R})\\e^{i\theta}&\mapsto&\begin{pmatrix}\cos(2\theta)&-\sin(2\theta)\\\sin(2\theta)&\cos(2\theta)\end{pmatrix}.\end{array}$$

Answer 2

As it happens $U(1)$ is an $n$-fold cover of itself for all positive integers $n$ by the mapping $e^{it}\mapsto e^{nit}$. Therefore it is also an $n$-fold covering of $SO(2)$ for all $n$.