Spin^h structure

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.

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:

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

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 equivalently a orientable real vector bundle of third rank.

• 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