Spinh structure

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

Definition

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

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

Properties

• Every spin and even every spin^c structure induces a spin^h structure. Reverse implications don't hold as the complex projective plane $ℂP^2$ and the Wu manifold $\mathrm{SU}(3)/\mathrm{SO}(3)$ show.
• If an orientable manifold $M$ has a spin^h 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 $H^4(M,{\mathbb{Z}}_2)\to H^4(M,\mathbb{Z})$.
• Every compact orientable smooth manifold with seven or less dimensions has a spin^h structure.
• In eight dimensions, there are infinitely many homotopy types of closed simply connected manifolds without spin^h structure.
• For a compact spin^h manifold $M$ of even dimension with either vanishing fourth Betti number $b_4(M)=\dim H^4(M,ℝ)$ or the first Pontrjagin class $p_1(E)\in H^4(M,\mathbb{Z})$ of its canonical principal $\mathrm{SO}(3)$-bundle $E\twoheadrightarrow M$ being torsion, twice its  genus $2\hat{A}(M)$ is integer.

The following properties hold more generally for the lift on the Lie group ${\mathrm{Spin}}^k(n):=(\mathrm{Spin}(n)\times \mathrm{Spin}(k))/{\mathbb{Z}}_2$, with the particular case $k=3$ giving:

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

• Spin^c structure

External links

• spinʰ structure on nLab