Spinh Structure

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

Definition

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

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

: \operatorname^\mathrm(M)
\cong ,\operatorname(1)\cong ,\mathbbP^\infty\cong ,K(\mathbb,4)_\mathbb

Due to the canonical projection $\operatorname^\mathrm(n)\rightarrow\operatorname(2)/\mathbb_2 \cong\operatorname(3)$, every spinÊ° structure induces a principal $\operatorname(3)$-bundle or equivalently 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 $\mathbbP^2$ and the Wu manifold $\operatorname(3)/\operatorname(3)$ show.
* If an orientable manifold $M$ has a spinʰ structur, then its fifth integral Stiefel–Whitney class $W_5(M) \in H^4(M,\mathbb)$ vanishes, hence is the image of the fourth ordinary Stiefel–Whitney class $w_4(M) \in H^4(M,\mathbb)$ under the canonical map $H^4(M,\mathbb_2)\rightarrow H^4(M,\mathbb)$.
* Every orientable smooth manifold with seven or less dimensions has a spinÊ° structure.
* In eight dimensions, there are infinitely many homotopy types of closed simply connected manifolds without spinÊ° structure.
* For a compact spinʰ manifold $M$ of even dimension with either vanishing fourth Betti number $b_4(M)=\dim H^4(M,\mathbb)$ or the first Pontrjagin class $p_1(E)\in H^4(M,\mathbb)$ of its canonical principal $\operatorname(3)$-bundle $E\twoheadrightarrow M$ being torsion, twice its  genus $2\widehat(M)$ is integer.Bär 1999, page 18

The following properties hold more generally for the lift on the Lie group $\operatorname^k(n) :=\left( \operatorname(n)\times\operatorname(k) \right)/\mathbb_2$, 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 spinÊ° manifold iff $N$ is a spinÊ° 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 spinÊ° 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

*
* {{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

* spinÊ° structure on ''n''Lab

References

Differential geometry