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## Spin cobordism of the equatorial S3 in S4

A physical universe in the S4 universe is a hypersurface of $S^4$.  We know that the equatorial $S^3$ has positive scalar curvature.  If we can deform our hypersurface smoothly through embeddings to the equatorial $S^3$ we can then use the Gromov-Lawson theorem that says that positive scalar curvature condition is preserved by this ‘spin cobordism’ and therefore $M$ has positive scalar curvature and hence no harmonic spinors by the Lichnerowicz theorem.  This is another way to show that our actual universe cannot have massless fermions.

There is a problem with the argument above unfortunately, which is that the Gromov-Lawson spin cobordism theorem assumes that the dimensions of the manifolds are at least 5.  Therefore we have to analyze the situation further.

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