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.{{Citation needed|date=March 2025}}

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 [[Principal U(1)-bundle| $\operatorname{BSp}(1)
\cong\operatorname{BSU}(2)$ ]], 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:{{Citation needed|date=March 2025}}

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

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.{{Citation needed|date=March 2025}}

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.Albanese & Milivojević 2021, Theorem 1.4.
In eight dimensions, there are infinitely many homotopy types of closed simply connected manifolds without spinʰ structure.Albanese & Milivojević 2021, Theorem 1.5.
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 Â genus $2\widehat{A}(M)$ is integer.Bär 1999, page 18

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.Albanese & Milivojević 2021, Proposition 3.6.
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.Albanese & Milivojević 2021, Proposition 3.7.
The following conditions are equivalent:Albanese & Milivojević 2021, Proposition 3.2.
$M$ 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.

Literature

{{cite journal |author=Christian Bär |date=1999 |title=Elliptic symbols |url=https://www.researchgate.net/publication/280877898 |language=en |volume=201 |issue=1 |periodical=Mathematische Nachrichten}}
{{cite journal |author=Michael Albanese und Aleksandar Milivojević |date=2021 |title=Spinʰ and further generalisations of spin |language=en |volume=164 |pages=104–174 |arxiv=2008.04934 |doi=10.1016/j.geomphys.2022.104709 |periodical=Journal of Geometry and Physics}}

External links

spinʰ structure on nLab

References

Category: Differential geometry