---
tag: RSPZ9TCFQ8
kind: concept
name: Spin structure
msc: [53C27, 57R15]
sources:
  - url: https://en.wikipedia.org/wiki/Spin_structure
same_as:
  wikidata: Q7577418
  wikipedia: Spin structure
---

## Summary

A spin structure on an oriented [[Riemannian manifold|7CBT4SM83Y]] lifts its bundle of oriented orthonormal frames to a principal bundle for the [[spin group|JJSXXJQGEH]].
The spin group is the double cover of the special orthogonal group.
It is the extra structure needed to define spinors and the Dirac operator on a manifold.
An oriented manifold admits a spin structure exactly when its second Stiefel–Whitney class vanishes.

## Explanations

### L1 human

Rotating an object a full turn brings it back, but some quantities, called spinors, change sign after one turn and return only after two turns.
To have spinors on a curved space, one must choose consistently, over the whole space, how to "unwind" rotations twice.
Such a consistent choice is a spin structure.
Every sphere of dimension at least $2$ has exactly one spin structure, while the complex projective plane has none.