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  "body": "\n## Summary\n\nA spin structure on an oriented [[Riemannian manifold|7CBT4SM83Y]] lifts its bundle of oriented orthonormal frames to a principal bundle for the spin group.\nThe spin group is the double cover of the special orthogonal group.\nIt is the extra structure needed to define spinors and the Dirac operator on a manifold.\nAn oriented manifold admits a spin structure exactly when its second Stiefel–Whitney class vanishes.\n\n## Explanations\n\n### L1 human\n\nRotating an object a full turn brings it back, but some quantities, called spinors, change sign after one turn and return only after two turns.\nTo have spinors on a curved space, one must choose consistently, over the whole space, how to \"unwind\" rotations twice.\nSuch a consistent choice is a spin structure.\nEvery sphere of dimension at least $2$ has exactly one spin structure, while the complex projective plane has none.\n",
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