# The relationship between the structure of spacetime and the existence of spinor field?

We all know that the existence of spinor fields implies that spacetime must be time-orientable. Thus that spacetime is time-orientable is a necessary condition for existence of spinor fields.

Geroch, R. (1968). Spinor structure of space-times in general relativity I, J. Math. Phys. 9, 1739-1744 proved this theorem: In a non-compact spacetime the existence of 4 continuous vector fields constituting a Minkowski tetrad at each point is necessary and suffienct for the existence of spinor fields.

My qustions:

1. For compact spacetimes, are there some necessary and suffienct conditions for existence of spinor fields?

2. For general $n$-dimensional Lorentzian manifolds, what's the necessary and suffienct conditions for existence of spinor fields?

• For a compact world manifold $X$, its Euler characteristic and the second Stiefel-Whitney class $w_2$ must be zero, and its first Pontryagin number must be multiple of 48.