Spinh structure

The article's lead section may need to be rewritten. Please review the lead layout guide and help improve the lead of this article if you can. (March 2025) (Learn how and when to remove this message)

In spin geometry, a spinʰ structure (or quaternionic spin structure) is a special classifying map that can exist for orientable manifolds. Such manifolds are called spinʰ manifolds. H stands for the quaternions, which are denoted ${\displaystyle \mathbb {H} }$ and appear in the definition of the underlying spinʰ group.

Let ${\displaystyle M}$ be a ${\displaystyle n}$-dimensional orientable manifold. Its tangent bundle ${\displaystyle TM}$ is described by a classifying map ${\displaystyle M\rightarrow \operatorname {BSO} (n)}$ into the classifying space ${\displaystyle \operatorname {BSO} (n)}$ of the special orthogonal group ${\displaystyle \operatorname {SO} (n)}$. It can factor over the map ${\displaystyle \operatorname {BSpin} ^{\mathrm {h} }(n)\rightarrow \operatorname {BSO} (n)}$ induced by the canonical projection ${\displaystyle \operatorname {Spin} ^{\mathrm {h} }(n)\twoheadrightarrow \operatorname {SO} (n)}$ on classifying spaces. In this case, the classifying map lifts to a continuous map ${\displaystyle M\rightarrow \operatorname {BSpin} ^{\mathrm {h} }(n)}$ into the classifying space ${\displaystyle \operatorname {BSpin} ^{\mathrm {h} }(n)}$ of the spinʰ group ${\displaystyle \operatorname {Spin} ^{\mathrm {h} }(n)}$, which is called spinʰ structure.^{[citation needed]}

Let ${\displaystyle \operatorname {BSpin} ^{\mathrm {h} }(M)}$ denote the set of spinʰ structures on ${\displaystyle M}$ up to homotopy. The first symplectic group ${\displaystyle \operatorname {Sp} (1)}$ is the second factor of the spinʰ group and using its classifying space ${\displaystyle \operatorname {BSp} (1)\cong \operatorname {BSU} (2)}$, which is the infinite quaternionic projective space ${\displaystyle \mathbb {H} P^{\infty }}$and a model of the rationalized Eilenberg–MacLane space ${\displaystyle K(\mathbb {Z} ,4)_{\mathbb {Q} }}$, there is a bijection:^{[citation needed]}

${\displaystyle \operatorname {BSpin} ^{\mathrm {h} }(M)\cong [M,\operatorname {BSp} (1)]\cong [M,\mathbb {H} P^{\infty }]\cong [M,K(\mathbb {Z} ,4)_{\mathbb {Q} }].}$

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

• Every spin and even every spinᶜ structure induces a spinʰ structure. Reverse implications don't hold as the complex projective plane ${\displaystyle \mathbb {C} P^{2}}$ and the Wu manifold ${\displaystyle \operatorname {SU} (3)/\operatorname {SO} (3)}$ show.
• If an orientable manifold ${\displaystyle M}$ has a spinʰ structur, then its fifth integral Stiefel–Whitney class ${\displaystyle W_{5}(M)\in H^{4}(M,\mathbb {Z} )}$ vanishes, hence is the image of the fourth ordinary Stiefel–Whitney class ${\displaystyle w_{4}(M)\in H^{4}(M,\mathbb {Z} )}$ under the canonical map ${\displaystyle H^{4}(M,\mathbb {Z} _{2})\rightarrow H^{4}(M,\mathbb {Z} )}$.
• Every orientable smooth manifold with seven or less dimensions has a spinʰ structure.^[1]
• In eight dimensions, there are infinitely many homotopy types of closed simply connected manifolds without spinʰ structure.^[2]
• For a compact spinʰ manifold ${\displaystyle M}$ of even dimension with either vanishing fourth Betti number ${\displaystyle b_{4}(M)=\dim H^{4}(M,\mathbb {R} )}$ or the first Pontrjagin class ${\displaystyle p_{1}(E)\in H^{4}(M,\mathbb {Z} )}$ of its canonical principal ${\displaystyle \operatorname {SO} (3)}$-bundle ${\displaystyle E\twoheadrightarrow M}$ being torsion, twice its  genus ${\displaystyle 2{\widehat {A}}(M)}$ is integer.^[3]

The following properties hold more generally for the lift on the Lie group ${\displaystyle \operatorname {Spin} ^{k}(n):=\left(\operatorname {Spin} (n)\times \operatorname {Spin} (k)\right)/\mathbb {Z} _{2}}$, with the particular case ${\displaystyle k=3}$ giving:

• If ${\displaystyle M\times N}$ is a spinʰ manifold, then ${\displaystyle M}$ and ${\displaystyle N}$ are spinʰ manifolds.^[4]
• If ${\displaystyle M}$ is a spin manifold, then ${\displaystyle M\times N}$ is a spinʰ manifold iff ${\displaystyle N}$ is a spinʰ manifold.^[4]
• If ${\displaystyle M}$ and ${\displaystyle N}$ are spinʰ manifolds of same dimension, then their connected sum ${\displaystyle M\#N}$ is a spinʰ manifold.^[5]
• The following conditions are equivalent:^[6]
  • ${\displaystyle M}$ is a spinʰ manifold.
  • There is a real vector bundle ${\displaystyle E\twoheadrightarrow M}$ of third rank, so that ${\displaystyle TM\oplus E}$ has a spin structure or equivalently ${\displaystyle w_{2}(TM\oplus E)=0}$.
  • ${\displaystyle M}$ can be immersed in a spin manifold with three dimensions more.
  • ${\displaystyle M}$ can be embedded in a spin manifold with three dimensions more.

• Spinᶜ structure

• Christian Bär (1999). "Elliptic symbols". Mathematische Nachrichten. 201 (1).
• Michael Albanese und Aleksandar Milivojević (2021). "Spinʰ and further generalisations of spin". Journal of Geometry and Physics. 164: 104–174. arXiv:2008.04934. doi:10.1016/j.geomphys.2022.104709.

• spinʰ structure on nLab