---
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.
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.