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We give a necessary and suffcient condition for almost-flat manifolds with cyclic holonomy to admit a Spin structure. Using this condition we find all 4-dimensional orientable almost- flat manifolds with cyclic holonomy that do not admit a Spin structure

We formulate a condition for the existence of a Spin C -structure on an oriented flat manifold M n with H 2 (M n , R) = 0. We prove that M n has a Spin Cstructure if and only if there exist a homomorphism ǫ : π 1 (M n ) → Spin C (n) suchλ n • ǫ = h, where h : π 1 (M n ) → SO(n) is a holonomy homomorphism and λ n : Spin C (n) → SO(n) is a standard homomorphism defined on page 2. As an application we shall prove that all cyclic Hantzsche -Wendt manifolds do not have the Spin C -structure.MSC2000: 53C27, 53C29, 20H15

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