Compact flat manifolds with non-vanishing Stiefel–Whitney classes

We construct a class of compact flat Riemannian manifolds M of dimension 2n+1 with the following properties: has holonomy group (Z2)n+1, is not a (non-trivial) flat toral extension of a compact flat manifold, e first Betti number of M is 0, and iefel–Whitney classes w2j(M) are non-zero for 0≤2j≤n. s in the spirit of Vasquezʹs second example which is in error (Vasquez, 1970). ch finite group Φ, there is a positive integer N(Φ) such that every flat manifold with holonomy group Φ has dimension higher than N(Φ), then M must be a non-trivial flat toral extension of a compact flat manifold. Vasquez pointed out that N((Z2)n)≥n or n−1 depending on n being even or odd. Our result shows that N((Z2)n)>2n−1, a much sharper result.