I have seen these lectures by Fredric Schuller that discuss the obstruction theory and the role of global geometric properties in admitting a spin structure.
See the video at 01:27:52
For example, it is said that electrons can live on a curved space like $S^2$ because, using the bundle language, it is a compact and orient-able base manifold and in particular its second class Stiefel–Whitney vanishes; which is a necessary and sufficient condition.
Now I am trying to imagine some physical intuition about the role of topology invariant constraints on the matter (physical) fields.
Is there any easy to understand geometry that does not allow the existence of electrons? I say that a Möbius shaped piece of metal should meet the Stiefel–Whitney condition, because it exists, but are there any shape to suggest which is not too much exotic to imagine and does not allow the existence of electrons on it due to the obstruction theory results?
Or the put it another way, which geometry does not allow the existence of matter?
Can you suggest a low dimensional and easy to understand geometry in which second class Stiefel–Whitney is not zero?