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Is the fundamental group of $Spin^c(2)$, the second complex spin group, also $\mathbb{Z}$? If so, how does one see this?

Just to avoid any confusion, my definition is: $$Spin^c(2) = (SO(2) \times \mathbb{S}^1)/\mathbb{Z}_2,$$ where $\mathbb{Z}_2$ is the group $\{(1, 1), (-1, -1)\}$.

I know that $Spin(2) \simeq SO(2) \simeq \mathbb{S}^1$, and I have the exact sequence

$$0 \rightarrow Spin(2) \rightarrow Spin^c(2) \rightarrow \mathbb{S}^1 \rightarrow 0,$$

where the first (non-trivial) map is given by inclusion and the second by the covering, so from this, going to homotopy, I get

$$0 \rightarrow \mathbb{Z} \rightarrow \pi_1(Spin^c(2)) \rightarrow \mathbb{Z} \rightarrow 0,$$

and this sequence splits, so it should be the case that $\pi_1(Spin^c(2)) = \mathbb{Z} \oplus \mathbb{Z}$. Is this correct? It doesn't seem to match other references that I have (Wikipedia, for instance).


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