Hantzsche-Wendt flat manifolds

The purpose of this paper is consider some results concerning the special class of the flat manifolds (closed, Riemannian with zero sectional curvature) of dimension n with holonomy groups (Z 2) . In dimension two it is the Klein bottle and in dimension three it is a flat manifold first consider by Hantzsche and Wendt in [4]. Hence we shall call this class of the HantzscheWendt flat manifolds (or the Hantzsche-Wendt Bieberbach groups). It was previous separately consider (see [9], [10], [11]) only in oriented case, where many interesting properties were observed. For example, there exist pairs of oriented isospectral Hantzsche-Wendt flat manifolds all not homeomorphic to each other (cf. [10]). Moreover they have I Q-homology of spheres (cf. [11]) and hence are geometrically formal (cf.[7]). The main result of a paper is proposition of a new, much general, definition of Hantzsche-Wendt flat manifolds which in oriented case is old one. Moreover we shall discuss some properties of it. For example the relations between the manifolds of different dimension, a I Q-homology and case with the first Betti number one. Finally we would like to mention that for us a class of flat manifolds (Bieberbach groups) consider in this article is still a little mysterious. There are many open problems and questions which we formulate at the end.