Spin structure

Definition

A spin structure on an oriented Riemannian manifold is a lift of its gluing cocycle to $SO(n)$, to a gluing cocycle to $Spin(n)$, the double cover of $SO(n)$.

Two spin structures are said to be equivalent if Fill this in later

The set of isomorphism classes of spin structures on a Riemannian manifold $M$ is denoted as $Spin(M)$.