We know that the universal cover of $SO(2)$ is $\mathbb{R}$ and the unitary group $U(2)$ has universal cover $SU(2) \times \mathbb R$, i,e. $\widetilde{U(2)}=SU(2) \times \mathbb R$ and $\widetilde{SO(2)}=\mathbb R$ . Then, so the space $ \widetilde{U(2)}/\widetilde{SO(2)}$ identify with which space? i.e., $$ \widetilde{U(2)}/\widetilde{SO(2)} = SU(2) \times \mathbb R/ \mathbb R = ???$$ Moreover, is what has $ \widetilde{U(2)/SO(2)}= \widetilde{U(2)}/\widetilde{SO(2)} ?$
Thank you in advance.