Q:

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: 

For compact spacetimes, are there some necessary and suffienct conditions for existence of spinor fields?
For general $n$-dimensional Lorentzian manifolds, what's the necessary and suffienct conditions for existence of spinor fields?

A:

Advanced Classical Field Theory (2009) by Giachetta, Mangiarotti, Sardanashvily remarks on p. 248:

A non-compact world manifold admits a Dirac spinor structure if and
  only if it is parallelizable.
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.

and gives the references

Geroch, R. (1968). Spinor structure of space-time in general relativity, J.
  Math. Phys. 9, 1739.
Wiston, G. (1974). Topics on space-time topology, Int. J. Theor. Phys. 11,
  341.

World manifolds are assumed orientable, simply connected and 4-dimensional, so you need to look at the references to see if it applies to arbitrary dimensions.

