Primitive Compact Flat Manifolds with Holonomy Group

From an important construction of Calabi (see [Ca], [Wo]), it follows that the compact Riemannian flat manifolds with first Betti number zero are the building blocks for all compact Riemannian flat manifolds. It is, therefore, of interest to construct families of such objects. These are often called primitive manifolds. Hantzsche and Wendt (1935) constructed the only existing 3-dimensional compact Riemannian flat manifold with first Betti number zero; this manifold has holonomy group Z2 ⊕ Z2. Cobb [Co] constructed a family of manifolds with these properties, for all dimensions n ≥ 3. In [RT] a rather larger family of primitive (Z2 ⊕ Z2)-manifolds was given. The goal in this paper is the classification, up to affine equivalence, of all primitive manifolds with holonomy group Z2 ⊕ Z2. We may notice that a similar project has been carried out in [RT2], where a full classification of 5-dimensional Bieberbach groups with holonomy group Z2 ⊕ Z2 was given. The classification is achieved by following a classical result of Charlap ([Ch1]), which reduces the problem to: