Spinc structure

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In spin geometry, a spinᶜ structure (or complex spin structure) is a special classifying map that can exist for orientable manifolds. Such manifolds are called spinᶜ manifolds. C stands for the complex numbers, which are denoted ${\displaystyle \mathbb {C} }$ and appear in the definition of the underlying spinᶜ group. In four dimensions, a spinᶜ structure defines two complex plane bundles, which can be used to describe negative and positive chirality of spinors, for example in the Dirac equation of relativistic quantum field theory. Another central application is Seiberg–Witten theory, which uses them to study 4-manifolds.

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 ${\displaystyle \operatorname {BSpin} ^{\mathrm {c} }(n)\rightarrow \operatorname {BSO} (n)}$ induced by the canonical projection ${\displaystyle \operatorname {Spin} ^{\mathrm {c} }(n)\twoheadrightarrow \operatorname {SO} (n)}$ on classifying spaces. In this case, the classifying map lifts to a continuous map ${\displaystyle M\rightarrow \operatorname {BSpin} ^{\mathrm {c} }(n)}$ into the classifying space ${\displaystyle \operatorname {BSpin} ^{\mathrm {c} }(n)}$ of the spinᶜ group ${\displaystyle \operatorname {Spin} ^{\mathrm {c} }(n)}$, which is called spinᶜ structure.^[1]

Let ${\displaystyle \operatorname {BSpin} ^{\mathrm {c} }(M)}$ denote the set of spinᶜ structures on ${\displaystyle M}$ up to homotopy. The first unitary group ${\displaystyle \operatorname {U} (1)}$ is the second factor of the spinᶜ group and using its classifying space ${\displaystyle \operatorname {BU} (1)\cong \operatorname {BSO} (2)}$, which is the infinite complex projective space ${\displaystyle \mathbb {C} P^{\infty }}$ and a model of the Eilenberg–MacLane space ${\displaystyle K(\mathbb {Z} ,2)}$, there is a bijection:^[2]

${\displaystyle \operatorname {BSpin} ^{\mathrm {c} }(M)\cong [M,\operatorname {BU} (1)]\cong [M,\mathbb {C} P^{\infty }]\cong [M,K(\mathbb {Z} ,2)]\cong H^{2}(M,\mathbb {Z} ).}$

Due to the canonical projection ${\displaystyle \operatorname {BSpin} ^{\mathrm {c} }(n)\rightarrow \operatorname {U} (1)/\mathbb {Z} _{2}\cong \operatorname {U} (1)}$, every spinᶜ structure induces a principal ${\displaystyle \operatorname {U} (1)}$-bundle or equvalently a complex line bundle.

• Every spin structure induces a canonical spinᶜ structure.^[3][4] The reverse implication doesn't hold as the complex projective plane ${\displaystyle \mathbb {C} P^{2}}$ shows.
• Every spinᶜ structure induces a canonical spinʰ structure. The reverse implication doesn't hold as the Wu manifold ${\displaystyle \operatorname {SU} (3)/\operatorname {SO} (3)}$ shows.^{[citation needed]}
• An orientable manifold ${\displaystyle M}$ has a spinᶜ structure iff its third integral Stiefel–Whitney class ${\displaystyle W_{3}(M)\in H^{2}(M,\mathbb {Z} )}$ vanishes, hence is the image of the second ordinary Stiefel–Whitney class ${\displaystyle w_{2}(M)\in H^{2}(M,\mathbb {Z} )}$ under the canonical map ${\displaystyle H^{2}(M,\mathbb {Z} _{2})\rightarrow H^{2}(M,\mathbb {Z} )}$.^[5]
• Every orientable smooth manifold with four or less dimensions has a spinᶜ structure.^[4]
• Every almost complex manifold has a spinᶜ structure.^[6][4]

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 ${\displaystyle k=2}$ giving:

• If ${\displaystyle M\times N}$ is a spinᶜ manifold, then ${\displaystyle M}$ and ${\displaystyle N}$ are spinᶜ manifolds.^[7]
• If ${\displaystyle M}$ is a spin manifold, then ${\displaystyle M\times N}$ is a spinᶜ manifold iff ${\displaystyle N}$ is a spinᶜ manifold.^[7]
• If ${\displaystyle M}$ and ${\displaystyle N}$ are spinᶜ manifolds of same dimension, then their connected sum ${\displaystyle M\#N}$ is a spinᶜ manifold.^[8]
• The following conditions are equivalent:^[9]
  • ${\displaystyle M}$ is a spinᶜ manifold.
  • There is a real plane bundle ${\displaystyle E\twoheadrightarrow M}$, 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 two dimensions more.
  • ${\displaystyle M}$ can be embedded in a spin manifold with two dimensions more.

• Spinʰ structure

• Blake Mellor (1995-09-18). "Spinᶜ manifolds" (PDF).
• "Stable complex and Spinᶜ-structures" (PDF).
• Liviu I. Nicolaescu. Notes on Seiberg-Witten Theory (PDF).
• 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.

• spinᶜ structure on nLab