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rotation groups in low dimensions:
see also
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
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:
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 $2 D_{2n}$:
On a spin 4-manifold, the $U(1)$-symmetry of the Seiberg-Witten equations? enhances to a $Pin(2)$-symmetry (Furuta 01). Because of this, $Pin(2)$-equivariance appears in Seiberg-Witten theory and Floer homology. See the references below.
rotation groups in low dimensions:
see also
$Pin(2)$-equivariant homotopy theory/equivariant cohomology theory
application to Seiberg-Witten theory and Floer homology:
Mikio Furuta, Monopole equation and the $11/8$-conjecture Mathematical Research Letters, Volume 8, Number 3, (2001).
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:
Last revised on June 14, 2019 at 06:23:06. See the history of this page for a list of all contributions to it.