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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 13th 2019

a minimum, for the moment just so as to record some references on $Pin(2)$-equivariant homotopy theory (as kindly pointed out by David Roberts)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeFeb 13th 2019

$\array{ 2 D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& Pin_-(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow \\ D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& O(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }$
• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 13th 2019

There’s a $Pin_{+}(2)$ group as well, I take it. Is either of them the default when $Pin(2)$ is mentioned?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 13th 2019

Yes, there is $Pin_+(2)$, obtained by changing the square of the reflection generator from $-1$ to +1. I focused on $Pin_-(2)$ just because that’s the choice that fits into the story of the finite subgroups of SU(2). But if anyone has the energy, it would be good to add discussion of $Pin_+(2)$, too.