Spinh structure

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 spin^h group ${\displaystyle \operatorname {Spin} ^{\mathrm {h} }(n)}$. Its homotopy class is called spin^h structure.^[2]

Assume ${\displaystyle M}$ has a spin^h structure. Let then ${\displaystyle \operatorname {Spin} ^{\mathrm {h} }(M)}$ denote the set of spin^h structures on ${\displaystyle M}$. The first symplectic group ${\displaystyle \operatorname {Sp} (1)}$ is the second factor of the spin^h 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 through its Postnikov tower projects onto the Eilenberg–MacLane space ${\displaystyle K(\mathbb {Z} ,4)}$, there is a map:^{[citation needed]}

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

The former isomorphism follows from the Puppe sequence for the fibration ${\displaystyle \mathbb {H} P^{\infty }\hookrightarrow \operatorname {BSpin} ^{\mathrm {h} }(n)\twoheadrightarrow \operatorname {BSO} (n)}$ (when applying ${\displaystyle [M,-]}$).^[3] Although this map is not a bijection in general, it is in special cases, for example for a 4-manifold ${\displaystyle M}$.

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

• Every spin and even every spin^c structure induces a spin^h 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.^[4]
• If an orientable manifold ${\displaystyle M}$ has a spin^h structure, then its fifth integral Stiefel–Whitney class ${\displaystyle W_{5}(M)\in H^{5}(M,\mathbb {Z} )}$ 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 compact orientable smooth manifold with seven or less dimensions has a spin^h structure.^[5]
• In eight dimensions, there are infinitely many homotopy types of closed simply connected manifolds without spin^h structure.^[6]
• For a compact spin^h 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.^[7]

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 spin^h manifold, then ${\displaystyle M}$ and ${\displaystyle N}$ are spin^h manifolds.^[8]
• If ${\displaystyle M}$ is a spin manifold, then ${\displaystyle M\times N}$ is a spin^h manifold iff ${\displaystyle N}$ is a spin^h manifold.^[8]
• If ${\displaystyle M}$ and ${\displaystyle N}$ are spin^h manifolds of same dimension, then their connected sum ${\displaystyle M\#N}$ is a spin^h manifold.^[9]
• The following conditions are equivalent:^[10]
  • ${\displaystyle M}$ is a spin^h 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.

The cohomology ring of the infinite classifying space ${\displaystyle \operatorname {BSpin} ^{\mathrm {h} }:=\lim _{n\rightarrow \infty }\operatorname {BSpin} ^{\mathrm {h} }(n)}$ with coefficients in ${\displaystyle \mathbb {Z} _{2}}$ can be expressed using Steenrod squares and Wu classes:^[11][12]

${\displaystyle H^{*}(\operatorname {BSpin} ^{\mathrm {h} },\mathbb {Z} _{2})\cong H^{*}(\operatorname {BSO} ,\mathbb {Z} _{2})/(\operatorname {Sq} ^{1}\nu _{2^{r}},r\geq 2).}$

• Spin^c structure

• Christian Bär (1999). "Elliptic symbols". Mathematische Nachrichten. 201 (1).
• Michael Albanese und Aleksandar Milivojević (2021). "Spin^h and further generalisations of spin". Journal of Geometry and Physics. 164: 104–174. arXiv:2008.04934. doi:10.1016/j.geomphys.2022.104709.
• H. Blaine Lawson (2023-01-23). "Spinʰ Manifolds". arXiv:2301.09683v1 [math.DG].
• Jiahao Hu (2023-12-08). "Invariants of Real Vector Bundles". arXiv:2310.05061 [math.AT].

• spinʰ structure on nLab