Problem about the notation $\text{Spin}^c(V)\cong \text{Spin}(V)\times _{\{\pm1\}} S^1$

Question

Lemma 2.6.1 of Morgan's book on Seiberg-Witten equations states that the group $\text{Spin}^c(V)$ is isomorphic to the group $\text{Spin}(V)\times _{\{\pm1\}} S^1$. The proof actually shows that $\text{Spin}^c(V)$ is isomorphic to the quotient group $\text{Spin}(V)\times S^1 / \{(\pm1 ,\pm1)\}$. Is $\text{Spin}(V)\times _{\{\pm1\}} S^1$ another notation for $\text{Spin}(V)\times S^1 / \{(\pm1 ,\pm1)\}$?

Answer 1

Generally I’ve seen $$A\times_CB := (A\times B)/C$$ where the relation is $$(ac,b) \sim (a,cb)$$

By $(\pm1,\pm1)$ do you mean $\pm(1,1)$? The former might be ambiguous. That is, are you including $(+1,-1)$ and $(-1,+1)$ or not? I think you shouldn't be including these, i.e. you should only be modding out by two elements, not four.

Once you write it as $\pm(1,1)$, you can say that, since $(1,1)$ is the identity of the group Spin$(V)\times S^1$, you can write it abstractly as the unit $1$. Thus the notation $\pm 1$ to mean $\pm(1,1)$.