# Spinc group

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In [spin geometry](/source/Spin_geometry), a **spinc group** (or **complex spin group**) is a [Lie group](/source/Lie_group) obtained by the [spin group](/source/Spin_group) through twisting with the first [unitary group](/source/Unitary_group). C stands for the [complex numbers](/source/Complex_number), which are denoted \mathbb{C}. An important application of spinc groups is for [spinc structures](/source/Spinc_structure), which are central for [Seiberg–Witten theory](/source/Seiberg%E2%80%93Witten_theory).

## Definition

The [spin group](/source/Spin_group) \operatorname{Spin}(n) is a [double cover](/source/Double_cover_(topology)) of the [special orthogonal group](/source/Special_orthogonal_group) \operatorname{SO}(n), hence \mathbb{Z}_2 acts on it with \operatorname{Spin}(n)/\Z_2\cong\operatorname{SO}(n). Furthermore, \mathbb{Z}_2 also acts on the first unitary group \operatorname{U}(1) through the [antipodal](/source/Antipodal_point) identification y\sim -y. The *spinc group* is then:[1][2][3][4]

- \operatorname{Spin}^\mathrm{c}(n) :=\left( \operatorname{Spin}(n)\times\operatorname{U}(1) \right)/\mathbb{Z}_2

with (x,y)\sim(-x,-y). It is also denoted \operatorname{Spin}^\mathbb{C}(n). Using the exceptional isomorphism \operatorname{Spin}(2) \cong\operatorname{U}(1), one also has \operatorname{Spin}^\mathrm{c}(n) =\operatorname{Spin}^2(n) with:

- \operatorname{Spin}^k(n) :=\left( \operatorname{Spin}(n)\times\operatorname{Spin}(k) \right)/\mathbb{Z}_2.

## Low-dimensional examples

- \operatorname{Spin}^\mathrm{c}(1) \cong\operatorname{U}(1) \cong\operatorname{SO}(2), induced by the isomorphism \operatorname{Spin}(1) \cong\operatorname{O}(1) \cong\mathbb{Z}_2
- \operatorname{Spin}^\mathrm{c}(3) \cong\operatorname{U}(2),[5] induced by the [exceptional isomorphism](/source/Exceptional_isomorphism) \operatorname{Spin}(3) \cong\operatorname{Sp}(1) \cong\operatorname{SU}(2). Since furthermore \operatorname{Spin}(2) \cong\operatorname{U}(1) \cong\operatorname{SO}(2), one also has \operatorname{Spin}^\mathrm{c}(3) \cong\operatorname{Spin}^\mathrm{h}(2).
- \operatorname{Spin}^\mathrm{c}(4) \cong\operatorname{U}(2)\times_{\operatorname{U}(1)}\operatorname{U}(2), induced by the exceptional isomorphism \operatorname{Spin}(4) \cong\operatorname{SU}(2)\times\operatorname{SU}(2)
- \operatorname{Spin}^\mathrm{c}(6) \rightarrow\operatorname{U}(4) is a double cover, induced by the exceptional isomorphism \operatorname{Spin}(6) \cong\operatorname{SU}(4)

## Properties

For all higher abelian [homotopy groups](/source/Homotopy_group), one has:

- \pi_k\operatorname{Spin}^\mathrm{c}(n) \cong\pi_k\operatorname{Spin}(n)\times\pi_k\operatorname{U}(1) \cong\pi_k\operatorname{SO}(n)

for k\geq 2.

## See also

- [Spinh group](/source/Spinh_group)

## Literature

- Lawson, Herbert Blaine Jr. & Michelsohn, Marie-Louise (1989). *Spin Geometry*. Vol. 38. Princeton Mathematical Series. Princeton: Princeton University Press. [doi:10.1515/9781400883912](https://doi.org/10.1515/9781400883912). ISBN 978-1-4008-8391-2.
- [Christian Bär](/source/Christian_B%C3%A4r) (1999). ["Elliptic symbols"](https://www.researchgate.net/publication/280877898). *Mathematische Nachrichten*. **201** (1)
- ["Stable complex and Spinc-structures"](https://ncatlab.org/nlab/files/StableComplexSpinC.pdf)
- Liviu I. Nicolaescu. [*Notes on Seiberg-Witten Theory*](https://www3.nd.edu/~lnicolae/new1.pdf)

## References

1. Lawson & Michelson 1989, Appendix D, Equation (D.1)

1. Bär 1999, page 14

1. *Stable complex and Spinc-structures*, section 2.1

1. Nicolaescu, page 30

1. Nicolaescu, Exercise 1.3.9

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