I have the following problem: Find a space $X$ path-connected, whose universal covering is contractible and is such that $\pi_{1}(X)=\mathbb{S}^{1}$, where $\mathbb{S}^{1}\subseteq\mathbb{C}$ and it is considered a group under multiplication.
A space $X$ that satisfies the above for a given group $G$ is usually called the space $K(G,1)$ (or Eilenberg-McLane space).
Now, since any element of $e^{i\theta}\in\mathbb{S}^{1}$ can be represented by the matrix \begin{equation*} \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \end{equation*}
then we can consider $\mathbb{S}^{1}\cong U(1)\cong SO(2)$, where $U(1)$ is the special unit group and $SO(2)$ is the group of matrices $2\times 2$ such that its determinant is equal to $1$.
Furthermore I know that for any given group $G$, there is a $2$-dimensional cell complex $X_{G}$ shuch that $\pi_{1}(X_{G})\cong G$, but how would you construct $X_{\mathbb{S}^{1}}$ in this case?
Furthermore, will the space $X_{\mathbb{S}^{1}}$ have a contractible space as a universal covering?
Any hint will be greatly appreciated.
