# Spinc structure

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In [spin geometry](/source/Spin_geometry), a **spinc structure** (or **complex spin structure**) is a generalization of a [spin structure](/source/Spin_structure). In mathematics, these are used to describe [spinor bundles](/source/Spinor_bundle) and [spinors](/source/Spinor), which in physics are used to describe [spin](/source/Spin_(physics)), an intrinsic [angular momentum](/source/Angular_momentum) of [particles](/source/Particle) after which they have been named. Since spinc structures also exist under weakened conditions, which might not allow spin structures, they provide a suitable alternative for such situations. [Orientable](/source/Orientability) manifolds with a spinc structure are called **spinc manifolds**.[1] C stands for the [complex numbers](/source/Complex_number), which are denoted \mathbb{C} and appear in the definition of the underlying [spinc group](/source/Spinc_group).

In four dimensions, a spinc structure defines two complex plane bundles, which can be used to describe negative and positive [chirality](/source/Chirality_(physics)) of [spinors](/source/Spinor), for example in the [Dirac equation](/source/Dirac_equation) of [relativistic quantum field theory](/source/Relativistic_quantum_field_theory). Another central application is [Seiberg–Witten theory](/source/Seiberg%E2%80%93Witten_theory), which uses them to study [4-manifolds](/source/4-manifold).

## Definition

Let M be a n-dimensional [orientable manifold](/source/Orientable_manifold). Its [tangent bundle](/source/Tangent_bundle) TM is described by a classifying map M\rightarrow\operatorname{BSO}(n) into the [classifying space](/source/Classifying_space_for_SO(n)) \operatorname{BSO}(n) of the [special orthogonal group](/source/Special_orthogonal_group) \operatorname{SO}(n). It can factor over the map \operatorname{BSpin}^\mathrm{c}(n)\rightarrow\operatorname{BSO}(n) induced by the canonical projection \operatorname{Spin}^\mathrm{c}(n)\twoheadrightarrow\operatorname{SO}(n) on [classifying spaces](/source/Classifying_space). In this case, the classifying map lifts to a continuous map M\rightarrow\operatorname{BSpin}^\mathrm{c}(n) into the classifying space \operatorname{BSpin}^\mathrm{c}(n) of the [spinc group](/source/Spinc_group) \operatorname{Spin}^\mathrm{c}(n). Its [homotopy](/source/Homotopy) class is called *spinc structure*.[2][3]

Assume M has a spinc structure. Let then \operatorname{Spin}^\mathrm{c}(M) denote the set of spinc structures on M. The first [unitary group](/source/Unitary_group) [\operatorname{U}(1)](/source/Principal_U(1)-bundle) is the second factor of the spinc group and using its [classifying space](/source/Classifying_space_for_U(n)) [\operatorname{BU}(1) \cong\operatorname{BSO}(2)](/source/Principal_U(1)-bundle), which is the infinite [complex projective space](/source/Complex_projective_space) [\mathbb{C}P^\infty](/source/Principal_U(1)-bundle) and a model of the [Eilenberg–MacLane space](/source/Eilenberg%E2%80%93MacLane_space) K(\mathbb{Z},2), there is a [bijection](/source/Bijection):[4]

- \operatorname{Spin}^\mathrm{c}(M) \cong[M,\operatorname{BU}(1)] \cong[M,\mathbb{C}P^\infty] \cong[M,K(\mathbb{Z},2)] \cong H^2(M,\mathbb{Z}).

The former isomorphism follows from the [Puppe sequence](/source/Puppe_sequence) for the fibration \mathbb{C}P^\infty\hookrightarrow\operatorname{BSpin}^\mathrm{c}(n)\twoheadrightarrow\operatorname{BSO}(n) (when applying [\[M,-\]](/source/Principal_U(1)-bundle)).[5]

Due to the canonical projection \operatorname{BSpin}^\mathrm{c}(n)\rightarrow\operatorname{U}(1)/\mathbb{Z}_2 \cong\operatorname{U}(1), every spinc structure induces a [principal \operatorname{U}(1)-bundle](/source/Principal_U(1)-bundle) or equivalently a complex [line bundle](/source/Line_bundle).

## Properties

- Every spin structure induces a canonical spinc structure.[6][7] The reverse implication doesn't hold as the [complex projective plane](/source/Complex_projective_plane) \mathbb{C}P^2 shows.
- Every spinc structure induces a canonical spinh structure. The reverse implication doesn't hold as the [Wu manifold](/source/Wu_manifold) \operatorname{SU}(3)/\operatorname{SO}(3) shows.[citation needed]
- An orientable manifold M has a spinc structure iff its third integral Stiefel–Whitney class W_3(M) \in H^2(M,\mathbb{Z}) vanishes, hence is the image of the second ordinary Stiefel–Whitney class w_2(M) \in H^2(M,\mathbb{Z}) under the canonical map H^2(M,\mathbb{Z}_2)\rightarrow H^2(M,\mathbb{Z}).[8][9]
- Every orientable smooth manifold with four or less dimensions has a spinc structure.[7]
- Every [almost complex manifold](/source/Almost_complex_manifold) has a spinc structure.[10][7]
- For a [compact](/source/Compact_space) spinc manifold M, for which a torsion class c\in H^2(M,\mathbb{Z}) with w_2(M)=c\operatorname{mod}2 exists and which has a Riemannian metric of overall positive scalar curvature, its [Â genus](/source/%C3%82_genus) vanishes, hence \widehat{A}(M)=0.[11]

The following properties hold more generally for the lift on the Lie group \operatorname{Spin}^k(n) :=\left( \operatorname{Spin}(n)\times\operatorname{Spin}(k) \right)/\mathbb{Z}_2, with the particular case k=2 giving:

- If M\times N is a spinc manifold, then M and N are spinc manifolds.[12]
- If M is a spin manifold, then M\times N is a spinc manifold iff N is a spinc manifold.[12]
- If M and N are spinc manifolds of same dimension, then their [connected sum](/source/Connected_sum) M\# N is a spinc manifold.[13]
- The following conditions are equivalent:[14] - M is a spinc manifold. - There is a real plane bundle E\twoheadrightarrow M, so that TM\oplus E has a spin structure or equivalently w_2(TM\oplus E) =0. - M can be immersed in a spin manifold with two dimensions more. - M can be embedded in a spin manifold with two dimensions more.

## Cohomology of infinite classifying space

The [cohomology ring](/source/Cohomology_ring) of the infinite classifying space \operatorname{BSpin}^\mathrm{c} :=\lim_{n\rightarrow\infty}\operatorname{BSpin}^\mathrm{c}(n) with coefficients in \mathbb{Z}_2 can be expressed using [Steenrod squares](/source/Steenrod_square) and [Wu classes](/source/Wu_class):[15][16]

- H^*(\operatorname{BSpin}^\mathrm{c},\mathbb{Z}_2) \cong H^*(\operatorname{BSO},\mathbb{Z}_2)/(\operatorname{Sq}^1\nu_{2^r},r\geq 1).

## See also

- [Spinh structure](/source/Spinh_structure)

## Literature

- Lawson, H. Blaine & Michelsohn, Marie-Louise (1990-02-21). *Spin Geometry*. [Princeton University Press](/source/Princeton_University_Press). ISBN 9780691085425.
- Blake Mellor (1995-09-18). ["Spinc manifolds"](https://www.maths.ed.ac.uk/~v1ranick/papers/mellor.pdf)
- ["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)
- Michael Albanese und Aleksandar Milivojević (2021). "Spinh and further generalisations of spin". *[Journal of Geometry and Physics](/source/Journal_of_Geometry_and_Physics)*. **164**: 104–174. [arXiv:2008.04934](https://arxiv.org/abs/2008.04934). [doi:10.1016/j.geomphys.2022.104709](https://doi.org/10.1016/j.geomphys.2022.104709)
- [H. Blaine Lawson](/source/H._Blaine_Lawson) (2023-01-23). "Spinʰ Manifolds". [arXiv:2301.09683v1](https://arxiv.org/abs/2301.09683v1)
- Jiahao Hu (2023-12-08). "Invariants of Real Vector Bundles". [arXiv:2310.05061](https://arxiv.org/abs/2310.05061)

## References

1. Lawson & Michelson 90, Definition D.3

1. Albanese & Milivojević 2021, Definition 3.1

1. *Stable complex and Spinc-structures*, Definition D.28

1. Mellor 1995, Theorem 5

1. Albanese & Milivojević 2021, p. 6

1. Mellor 1995, Theorem 2

1. Nicolaescu, Example 1.3.16

1. Lawson & Michelson 90, Theorem D.2 und Corollary D.4

1. *Stable complex and Spinc-structures*, Proposition D.31

1. Mellor 1995, Theorem 3

1. Lawson & Michelson 90, Corollary D.16

1. Albanese & Milivojević 2021, Proposition 3.6.

1. Albanese & Milivojević 2021, Proposition 3.7.

1. Albanese & Milivojević 2021, Proposition 3.2.

1. Lawson 2023, p. 8

1. Hu 2023, Rem. 4.30

## External links

- on [*n*Lab](/source/NLab)

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Adapted from the Wikipedia article [Spinc structure](https://en.wikipedia.org/wiki/Spinc_structure) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Spinc_structure?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.