Topology of Spacetime: Can Singularities and Fermions Co-exist?

[Narasoma]

I watched this video : https://www.youtube.com/watch?v=sOiifkFYck4
Here, the lecturer said that if someone wants a spacetime which contains spin structure (physically equal to the existence of fermions, CMIIW) should topologically ℝ×Σ, where Σ is the Cauchy surface.

Is that true? If so, then singularities and fermions can't co-exist in the same universe, which lead us to a paradox, right?
Anybody could give me some papers related to this topic please?

 

[PeterDonis]

Is that true?

I'm not sure; I'm not familiar with any papers on the subject. However, it seems plausible.

If so, then singularities and fermions can't co-exist in the same universe

Why not? There are spacetimes with singularities that have topology $R \times \Sigma$, where $\Sigma$ is a 3-manifold. The simplest example is FRW spacetime.

which lead us to a paradox, right?

o

If it's true (and I'm not sure it is, see above), it is only a paradox if you think our actual universe contains singularities; I take it we agree that it contains fermions. But we don't know that that's the case.