# Spinh structure

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In [spin geometry](/source/Spin_geometry), a **spinh structure** (or **quaternionic 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 spinh 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 spinh structures are called **spinh manifolds**.[1] H stands for the [quaternions](/source/Quaternion), which are denoted \mathbb{H} and appear in the definition of the underlying [spinh group](/source/Spinh_group).

## 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{h}(n)\rightarrow\operatorname{BSO}(n) induced by the canonical projection \operatorname{Spin}^\mathrm{h}(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{h}(n) into the classifying space \operatorname{BSpin}^\mathrm{h}(n) of the [spinh group](/source/Spinh_group) \operatorname{Spin}^\mathrm{h}(n). Its [homotopy](/source/Homotopy) class is called *spinh structure*.[2]

Assume M has a spinh structure. Let then \operatorname{Spin}^\mathrm{h}(M) denote the set of spinh structures on M. The first symplectic group [\operatorname{Sp}(1)](/source/Principal_U(1)-bundle) is the second factor of the spinh group and using its [classifying space](/source/Classifying_space_for_SU(n)) [\operatorname{BSp}(1) \cong\operatorname{BSU}(2)](/source/Principal_U(1)-bundle), which is the infinite [quaternionic projective space](/source/Quaternionic_projective_space) [\mathbb{H}P^\infty](/source/Principal_U(1)-bundle) and through its [Postnikov tower](/source/Postnikov_tower) projects onto the [Eilenberg–MacLane space](/source/Eilenberg%E2%80%93MacLane_space) K(\mathbb{Z},4), there is a map:[citation needed]

- \operatorname{Spin}^\mathrm{h}(M) \cong[M,\operatorname{BSp}(1)] \cong[M,\mathbb{H}P^\infty] \rightarrow[M,K(\mathbb{Z},4)] \cong H^4(M,\mathbb{Z}).

The former isomorphism follows from the [Puppe sequence](/source/Puppe_sequence) for the fibration \mathbb{H}P^\infty\hookrightarrow\operatorname{BSpin}^\mathrm{h}(n)\twoheadrightarrow\operatorname{BSO}(n) (when applying [\[M,-\]](/source/Principal_U(1)-bundle)).[3] Although this map is not a bijection in general, it is in special cases, for example for a [4-manifold](/source/4-manifold) M.

Due to the canonical projection \operatorname{BSpin}^\mathrm{h}(n)\rightarrow\operatorname{SU}(2)/\mathbb{Z}_2 \cong\operatorname{SO}(3), every spinh structure induces a principal \operatorname{SO}(3)-bundle or equivalently a orientable real vector bundle of third rank.[citation needed]

## Properties

- Every spin and even every [spinc structure](/source/Spinc_structure) induces a spinh structure. Reverse implications don't hold as the [complex projective plane](/source/Complex_projective_plane) \mathbb{C}P^2 and the [Wu manifold](/source/Wu_manifold) \operatorname{SU}(3)/\operatorname{SO}(3) show.[4]
- If an orientable manifold M has a spinh structure, then its fifth integral Stiefel–Whitney class W_5(M) \in H^5(M,\mathbb{Z}) vanishes, hence is the image of the fourth ordinary Stiefel–Whitney class w_4(M) \in H^4(M,\mathbb{Z}) under the canonical map H^4(M,\mathbb{Z}_2)\rightarrow H^4(M,\mathbb{Z}).
- Every compact orientable smooth manifold with seven or less dimensions has a spinh structure.[5]
- In eight dimensions, there are infinitely many [homotopy types](/source/Homotopy_type) of [closed](/source/Closed_manifold) [simply connected](/source/Simply_connected_space) manifolds without spinh structure.[6]
- For a [compact](/source/Compact_space) spinh manifold M of even dimension with either vanishing fourth [Betti number](/source/Betti_number) b_4(M)=\dim H^4(M,\mathbb{R}) or the first [Pontrjagin class](/source/Pontryagin_class) p_1(E)\in H^4(M,\mathbb{Z}) of its canonical principal \operatorname{SO}(3)-bundle E\twoheadrightarrow M being torsion, twice its [Â genus](/source/%C3%82_genus) 2\widehat{A}(M) is integer.[7]

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=3 giving:

- If M\times N is a spinh manifold, then M and N are spinh manifolds.[8]
- If M is a spin manifold, then M\times N is a spinh manifold iff N is a spinh manifold.[8]
- If M and N are spinh manifolds of same dimension, then their [connected sum](/source/Connected_sum) M\# N is a spinh manifold.[9]
- The following conditions are equivalent:[10] - M is a spinh manifold. - There is a real vector bundle E\twoheadrightarrow M of third rank, 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 three dimensions more. - M can be embedded in a spin manifold with three dimensions more.

## Cohomology of infinite classifying space

The [cohomology ring](/source/Cohomology_ring) of the infinite classifying space \operatorname{BSpin}^\mathrm{h} :=\lim_{n\rightarrow\infty}\operatorname{BSpin}^\mathrm{h}(n) with coefficients in \mathbb{Z}_2 can be expressed using [Steenrod squares](/source/Steenrod_square) and [Wu classes](/source/Wu_class):[11][12]

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

## See also

- [Spinc structure](/source/Spinc_structure)

## Literature

- [Christian Bär](/source/Christian_B%C3%A4r) (1999). ["Elliptic symbols"](https://www.researchgate.net/publication/280877898). *Mathematische Nachrichten*. **201** (1)
- 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)

## External links

- [spinʰ structure](/source/Nlab:spin%CA%B0%2Bstructure) on [*n*Lab](/source/NLab)

## References

1. Hu 2023, Def. 4.3

1. Albanese & Milivojević 2021, Definition 3.1

1. Albanese & Milivojević 2021, p. 5

1. Lawson 2023, p. 3

1. Albanese & Milivojević 2021, Theorem 1.4.

1. Albanese & Milivojević 2021, Theorem 1.5.

1. Bär 1999, page 18

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, Thrm. 4.29

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Adapted from the Wikipedia article [Spinh structure](https://en.wikipedia.org/wiki/Spinh_structure) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Spinh_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.