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Almost‐flat manifolds were defined by Gromov as a natural generalization of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost‐flat manifold is determined by the canonical orthogonal representation of its fundamental group. Utilizing this, we classify the spin structures on all four‐dimensional almost‐flat manifolds that are not flat. Out of 127 orientable families, we show that there are exactly 15 that are non‐spin, the rest are, in fact, parallelizable.

CLASSIFICATION OF SPIN STRUCTURES ON FOUR‐DIMENSIONAL ALMOST‐FLAT MANIFOLDS