Math 2.0

Differential geometry Definition

Spin structure

Builds on Riemannian manifoldOrientation of a manifoldFrame bundlePrincipal bundleStiefel–Whitney class

Definition

Let (M,g)(M, g) be an oriented Riemannian manifold of dimension n≥2n \ge 2. Let PSO(M)P_{SO}(M) be its bundle of oriented orthonormal frames, a principal SO(n)SO(n)-bundle (see frame bundle). Let ρ ⁣:Spin⁡(n)→SO(n)\rho \colon \operatorname{Spin}(n) \to SO(n) be the double covering homomorphism from the spin group. A spin structure on (M,g)(M, g) is a pair (P,ξ)(P, \xi) such that:

  1. PP is a principal Spin⁡(n)\operatorname{Spin}(n)-bundle over MM.
  2. ξ ⁣:P→PSO(M)\xi \colon P \to P_{SO}(M) is a smooth map that commutes with the projections to MM.
  3. ξ(p⋅a)=ξ(p)⋅ρ(a)\xi(p \cdot a) = \xi(p) \cdot \rho(a) for all p∈Pp \in P and a∈Spin⁡(n)a \in \operatorname{Spin}(n).

Two spin structures (P,ξ)(P, \xi) and (P′,ξ′)(P', \xi') are equivalent if there is a Spin⁡(n)\operatorname{Spin}(n)-equivariant bundle isomorphism ϕ ⁣:P→P′\phi \colon P \to P' with ξ′∘ϕ=ξ\xi' \circ \phi = \xi.

Remarks

Clauses 1 to 3 imply that ξ\xi is a two-sheeted covering map. An oriented manifold admits a spin structure if and only if its second Stiefel–Whitney class w2(TM)w_2(TM) vanishes. In that case the equivalence classes of spin structures are in bijection with H1(M;ℤ/2)H^1(M; \Z/2), after one class is fixed. Existence and the set of classes do not depend on the choice of the metric gg.

Example: the sphere SnS^n with n≥2n \ge 2 has w2=0w_2 = 0, because its tangent bundle becomes trivial after adding a trivial line bundle. Since H1(Sn;ℤ/2)=0H^1(S^n; \Z/2) = 0, it has exactly one spin structure up to equivalence. For SnS^n it is Spin⁡(n+1)→SO(n+1)=PSO(Sn)\operatorname{Spin}(n+1) \to SO(n+1) = P_{SO}(S^n).

Non-example: the complex projective plane ℂP2\C P^2 is oriented but w2(TℂP2)≠0w_2(T\C P^2) \ne 0, so no pair (P,ξ)(P, \xi) satisfies the clauses. A manifold with a spin structure is called a spin manifold.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Frame bundle, concept, no evidence levelFrame bundleconceptOrientation of a manifold, concept, no evidence levelOrientation of a mani…conceptPrincipal bundle, concept, no evidence levelPrincipal bundleconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptStiefel–Whitney class, concept, no evidence levelStiefel–Whitney classconceptSpin structure, definition, no evidence levelSpin structuredefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Frame bundle concept
  2. Orientation of a manifold concept
  3. Principal bundle concept
  4. Riemannian manifold concept
  5. Stiefel–Whitney class concept
  6. Spin structure definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Spin_structure
About this page
TagQYZNMEM1ZZ
URIm2://QYZNMEM1ZZ
Kinddefinition
WritingW0 stub
MSC53C27, 57R15
DefinesSpin structure
UsesRiemannian manifold, Orientation of a manifold, Frame bundle, Principal bundle, Stiefel–Whitney class
Changed2 November 2026

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