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It is a theorem that spin structures on a spacetime $M$ exist iff the second Stiefel-Whitney class $w_2(M)=H^2(M, \mathbb{Z}_2)$ vanishes. I find this confusing for two reasons.

First, it implies that merely knowing that, say, electrons exist gives us some information about the global topology of spacetime. In particular, this sounds like it might violate a naive conception of causality, since we are learning about parts of the universe beyond the particle horizon. In other words, it doesn't seem reasonable that the existence of electrons (fairly well-localized entities) should somehow be contingent on global features of the universe.

Second, it leads to the following thought experiment. If I were able to change the topology of space in my backyard so as to make $w_2(M)\neq 0$, I would erase spinors from existence and in so doing be able to send out superluminal signals (unless doing this would send out some sort of "shockwave" at or below the speed of light which would destroy spinor fields as it moved along). Therefore, such modifications should not be possible, and this tells us something about what kind of topology changes a theory of quantum gravity should allow. But this intuitively feels like much too strong a conclusion to be derivable from merely sitting in my chair contemplating spinors.

I feel like I'm either missing something very obvious or am making some mistake in my reasoning by being too sloppy, but I can't seem to put my finger on where the error lies. Can someone help me out?

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    $\begingroup$ I was gonna bring up how you could have pin structures instead of spin structures for topologically non-trivial spacetimes, which look remarkably like spin structures locally but not globally, but it turns out that even pin structures have topological obstructions. $\endgroup$
    – Slereah
    Jun 14 at 15:09
  • $\begingroup$ Interesting. On a different note, I was thinking that maybe the fact that in QFT only quadratic expressions in fermion fields should be observable (due to anticommutation at spacelike separation) has some role to play, but I can't quite see what. (Perhaps we can replace all occurences of fermions in our theories with their squares, and these squares would no longer be spinors? Probably not.) $\endgroup$
    – J_P
    Jun 14 at 17:30
  • $\begingroup$ Well I suspect that the topological obstruction may disappear if you go for a cover of that manifold, but I'm not sure. $\endgroup$
    – Slereah
    Jun 14 at 18:17
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    $\begingroup$ It looks like there is a notion called the projective spin structure that can be defined on any (orientable) manifold, it is possible that by bumping it to projective pin group that may cover all manifolds $\endgroup$
    – Slereah
    Jun 14 at 20:06
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    $\begingroup$ Not a lot on the topic but you can try here : arxiv.org/pdf/math/0402329.pdf $\endgroup$
    – Slereah
    Jun 14 at 20:13

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