Spinh 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 $\mathbb {H}$ and appear in the definition of the underlying spinʰ group.

Definition

Let $M$ be a $n$-dimensional orientable manifold. Its tangent bundle $TM$ is described by a classifying map $M\rightarrow \operatorname {BSO} (n)$ into the classifying space $\operatorname {BSO} (n)$ of the 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. 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 $\operatorname {Spin} ^{\mathrm {h} }(n)$, which is called spinʰ structure.

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

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

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

Properties

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

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.
• If $M$ is a spin manifold, then $M\times N$ is a spinʰ manifold iff $N$ is a spinʰ manifold.
• If $M$ and $N$ are spinʰ manifolds of same dimension, then their connected sum $M\#N$ is a spinʰ manifold.
• The following conditions are equivalent:
  • $M$ 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.

See also

• Spinᶜ structure

Literature

• 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.

External links

• spinʰ structure on nLab