Contents

group theory

spin geometry

string geometry

# Contents

## Idea

A Pin group in dimension 2.

## Properties

Let $\mathbb{H} \simeq \mathbb{C} \oplus j \mathbb{C}$ be the quaternions realized as the Cayley-Dickson double of the complex numbers, and identify the circle group

$SO(2) \simeq S\big( \mathbb{C}\big) \hookrightarrow \mathbb{H}$

with the unit circle in $\mathbb{C} \hookrightarrow \mathbb{H}$ this way, with group structure now thought of as given by multiplication of quaternions. Then the Pin group $Pin_-(2)$ is isomorphic to the subgroup of the group of units $\mathbb{H}^\times$ of the quaternions which consists of this copy of SO(2), together with the multiplies of the imaginary quaternion $j$ with this copy:

$Pin_-(2) \;\simeq\; S\big( \mathbb{C}\big) \;\cup\; j \cdot S\big( \mathbb{C}\big) \;\subset\; \mathbb{H}^\times \,.$

This is in fact inside the unit sphere $S(\mathbb{H}) \simeq$ Spin(3).

By the classification of the finite subgroups of Spin(3) this means that the restriction of $Pin_-(2)$ to the inclusion of the dihedral group $D_{2n}$ into the orthogonal group $O(2)$ is the binary dihedral group $2 D_{2n}$:

$\array{ 2 D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& Pin_-(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow \\ D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& O(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }$

## References

$Pin(2)$-equivariant homotopy theory/equivariant cohomology theory

application to Seiberg-Witten theory and Floer homology:

• Ciprian Manolescu, $Pin(2)$-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture (arXiv:1303.2354)

• Matthew Stoffregen, $Pin(2)$-equivariant Seiberg-Witten Floer homology of Seifert fibrations (arXiv:1505.03234)

• Matthew Stoffregen, A Remark on $Pin(2)$-Equivariant Floer Homology, Michigan Math. J. Volume 66, Issue 4 (2017), 867-884 (euclid:1508810818)

$Pin(2)$-equivariant KO-theory:

• Jianfeng Lin, Pin(2)-equivariant KO-theory and intersection forms of spin four-manifolds, Algebr. Geom. Topol. 15 (2015) 863-902 (arXiv:1401.3264)

Last revised on March 22, 2019 at 09:14:24. See the history of this page for a list of all contributions to it.