Orientability, Spin Structures, and Global Obstructions

Local orthonormal frames always exist, but they need not glue into the global structure required by orientation or spinors. The obstructions occur in stages. The first Stiefel–Whitney class tests orientation; Lorentzian time orientation is a separate choice; after both are fixed, the second Stiefel–Whitney class tests whether the oriented orthonormal frame bundle lifts through the double cover by the spin group. Even when a lift exists, it need not be unique.

Required background. Tensor Fields and Natural Operations supplies tangent and tensor bundles as geometric examples; Vector Bundles and Transition Functions supplies frames and transition maps. Exterior Algebra and Differential Forms supplies top forms, while Orientation, Densities, and Volume Elements develops orientation and integration.

Helpful background. Calculus and Linear Algebra supplies orthogonal groups and double covers; The Riemann Curvature Tensor motivates orthonormal frame connections.

Orientation and time orientation are different

For a real rank-$n$ vector bundle $E$, its first Stiefel–Whitney class

$$w_1(E)\in H^1(M;\mathbb Z_2)$$

vanishes exactly when the structure group can be reduced from $O(n)$ to $SO(n)$. For $E=TM$, this is ordinary orientability. In transition-function language, one can choose local frames whose overlap matrices all have positive determinant.

A Lorentzian metric adds another disconnected choice. A Lorentzian manifold is time-orientable when its timelike cones admit a continuous choice of future component, equivalently when it possesses a continuous nowhere-zero timelike vector field. This condition concerns the timelike cone bundle, not the sign of coordinate Jacobians. A spacetime can be orientable but not time-orientable, or time-orientable but not orientable. Choosing both reduces the orthonormal frame bundle to the proper, orthochronous group $SO^+(1,n-1)$.

These conditions should also be separated from parallelizability. A global frame trivializes $TM$ and is much stronger than orientation. Vanishing $w_1$ gives a compatible orientation, not $n$ independent vector fields.

The spin-lift obstruction

For spacetime dimension $n\ge3$, the group $\operatorname{Spin}^+(1,n-1)$ double-covers $SO^+(1,n-1)$. A spin structure on an oriented, time-oriented Lorentzian manifold is a principal $\operatorname{Spin}^+(1,n-1)$ bundle together with a two-to-one equivariant map onto the proper orthochronous orthonormal frame bundle. Choose a good cover, whose nonempty finite overlaps are contractible. Then every transition map on a double overlap lifts. On triple overlaps, the lifted maps may multiply to $+1$ or $-1$. Those signs form a Čech two-cocycle.

The resulting cohomology class is $w_2(TM)\in H^2(M;\mathbb Z_2)$. The fundamental criterion is

$$M\text{ admits a spin structure} \quad\Longleftrightarrow\quad w_2(TM)=0,$$

provided the orientation and, in Lorentzian signature, time orientation needed to define the connected frame bundle have already been chosen. Wernli constructs the lifted cocycle and states the Riemannian obstruction theorem in Wernli 2019, §2.3.1, Theorem 2.3.15, pp. 60–66. To relate it to the Lorentzian statement, choose a global future unit timelike field $u$. It splits $TM$ as the trivial timelike line plus the positive-definite bundle $u^\perp$. The Whitney product rule gives $w_2(TM)=w_2(u^\perp)$, and the Lorentz frame group retracts to the spatial rotation group. Thus the same obstruction governs the time-oriented Lorentzian lift.

The proof architecture is concrete. Any local lifts fail the triple-overlap condition by central signs. Changing the lifts changes those signs by a Čech coboundary, so their class is well defined. If a spin lift exists, its cocycle is exact and $w_2=0$. Conversely, when the class vanishes, one modifies the local lifts by signs so that the genuine cocycle condition holds, then glues the spin bundle.

Existence does not imply uniqueness

For a fixed oriented frame bundle, when spin structures exist their isomorphism classes form a torsor for $H^1(M;\mathbb Z_2)$. A torsor has differences but no preferred zero: twisting one lift by a $\mathbb Z_2$ one-cocycle gives another, and every other lift arises this way Wernli 2019, §2.3.1, Theorem 2.3.16, p. 67. On a simply connected manifold this group vanishes, so the spin structure is unique up to isomorphism. A simple Riemannian illustration is the circle, for which

$$H^1(S^1;\mathbb Z_2)\cong\mathbb Z_2,$$

so there are two spin structures. To interpret them through transport, give the circle the metric $d\theta^2$, with $\theta\sim\theta+2\pi$, and use its global parallel unit tangent $\partial_\theta$. The Levi–Civita connection has zero local connection form in this frame. Its spin lift therefore has holonomy $+1$ or $-1$, determined by the spin transition across the angular seam. In this frame the corresponding spinor components are periodic or antiperiodic. These signs describe this lifted Levi–Civita connection; a spinor bundle with an additional gauge connection has additional transport data. One of the two spin structures extends over a disk and the other does not; the distinction is global even though the local Clifford algebra is identical. Clifford Algebras and Spin Structures gives the analogous flat spacetime example.

Spin structure is not the same as a spinor field. The former permits construction of a spinor bundle; a spinor field is a section of that associated bundle. Nor does a spin structure imply a global orthonormal tetrad in arbitrary dimension. Hatcher explains the roles of $w_1$, $w_2$, and related obstruction classes in Hatcher 2017, §3.3, pp. 98–106.

Lorentzian scope and alternatives

If spacetime orientation or time orientation is absent, the relevant lifting question involves a larger orthogonal group and Pin-type structures rather than the spin structure defined above. A $\operatorname{Spin}^c$ structure is another, weaker lift involving a $U(1)$ factor; it should not be called an ordinary spin structure. These alternatives have different obstruction classes.

For a connected, noncompact four-dimensional spacetime without boundary, with the temporal and spatial orientations needed to define its restricted Lorentz frame bundle, Geroch’s theorem states that admitting the spin structure defined above is equivalent to admitting a global orthonormal tetrad field Geroch 1968, §II, pp. 1742–1743. The proof extends a section of the spin bundle over successive simplices. The vanishing of the first two homotopy groups of $SL(2,\mathbb C)$ permits extension through dimension three; a noncompact exhaustion removes the remaining fourth-degree obstruction. Projecting that section gives the tetrad. Noncompactness and four dimensions are essential hypotheses, not incidental physical assumptions. The result should not be generalized to manifolds with boundary, arbitrary vector bundles, or other dimensions without a separate argument. This page establishes topological existence and multiplicity only, not a Dirac operator, spinor dynamics, or quantization.

Exercise: spin structures on the circle

Use $H^1(S^1;\mathbb Z_2)$ to count spin structures on $S^1$. Explain why cutting the circle to an interval makes the two choices locally indistinguishable and how regluing distinguishes them.

Solution

Since $H^1(S^1;\mathbb Z_2)\cong\mathbb Z_2$, the torsor contains two isomorphism classes. An interval is contractible, so its first cohomology vanishes and every spin lift there is equivalent. Relative to the parallel tangent frame for $d\theta^2$, endpoint regluing uses either central element of the double cover. Thus the spinor components obey $\psi(\theta+2\pi)=\psi(\theta)$ or $\psi(\theta+2\pi)=-\psi(\theta)$. The lifted Levi–Civita connection is locally zero in this frame, so its parallel transport around the loop gives precisely the gluing sign. The local equations on the cut interval agree; their global boundary conditions differ. This conclusion does not assign a holonomy to an arbitrary additional connection.

References

• Geroch, Robert. “Spinor Structure of Space-Times in General Relativity. I.” Journal of Mathematical Physics 9 (1968): 1739–1744. DOI.
• Hatcher, Allen. Vector Bundles and K-Theory. Version 2.2, 2017. §3.3, pp. 98–106.
• Wernli, Konstantin. Lecture Notes on Spin Geometry. arXiv:1911.09766, 2019. §2.3.1, pp. 60–67.