nLab
Pin(2)

Contents

Context

Group Theory

Spin geometry

Contents

Idea

A Pin group in dimension 2.

Properties

Let j\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)S() 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)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 jj with this copy:

Pin (2)S()jS() ×. 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()S(\mathbb{H}) \simeq Spin(3).

By the classification of the finite subgroups of Spin(3) this means that the restriction of Pin (2)Pin_-(2) to the inclusion of the dihedral group D 2nD_{2n} into the orthogonal group O(2)O(2) is the binary dihedral group 2D 2n2 D_{2n}:

2D 2n AA Pin (2) AA Spin(3) (pb) (pb) D 2n AA O(2) AA SO(3) \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) }

Appearance in Seiberg-Witten theory and Floer homology

On a spin 4-manifold, the U(1)U(1)-symmetry of the Seiberg-Witten equations? enhances to a Pin(2)Pin(2)-symmetry (Furuta 01). Because of this, Pin(2)Pin(2)-equivariance appears in Seiberg-Witten theory and Floer homology. See the references below.

rotation groups in low dimensions:

sp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)Pin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)SO(8)
SO(9)Spin(9)
\vdots\vdots
SO(16)Spin(16)SemiSpin(16)
SO(32)Spin(32)SemiSpin(32)

see also

References

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

application to Seiberg-Witten theory and Floer homology:

  • Mikio Furuta, Monopole equation and the 11/811/8-conjecture Mathematical Research Letters, Volume 8, Number 3, (2001).

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

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

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

Pin(2)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 June 14, 2019 at 02:23:06. See the history of this page for a list of all contributions to it.