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## The universe is a compact spin manifold: in fact it is a scaled 4-sphere

Although Dirac discovered the matrices he needed so the square of a first order operator would be the wave operator for the Klein-Gordon equation in 1928, the algebraic structure and the right definition for compact manifolds was pursued by Atiyah, Singer and others.  One simple consequence of the Atiyah-Singer analysis is that the topological obstruction is the second Stiefel-Whitney class of the tangent bundle:  the manifold is spin if and only if the second Stiefel-Whitney number (which for complex vector bundles is the first Chern class mod 2) is zero.  This is automatic for a 4-sphere which has no 2-cohomology.

The universe is actually a closed quaternionic projective line, and these projective spaces of all dimensions are spin by results in Lawson-Michelson book.  The spin condition is not a very strong restriction of the shape of the universe and does in no way determine the shape.  In this way one could look at Dirac’s contribution to determining the shape of the universe: the existence of a spin structure which allows the Clifford module of spinors to be constructed as well as the Dirac matrices.

From Lawson-Michelson:  Suppose $X$ is a spin manifold with a spin structure on its tangent bundle.  Let $S$ be any spinor bundle associated to $T(X)$.  Then $S$ is a bundle of modules over $Cl(X)$, carries a canonical riemannian connection.  The Dirac operator in this case was first written down by Atiyah and Singer in their work on the Index Theorem.  Finding the operator was a major accomplishment so Lawson-Michelson call it the Atiyah-Singer operator.