Flat manifolds with prescribed first Betti number admitting Anosov diffeomorphisms (Q1601790)

The author, basing on the paper by \textit{H. L. Porteous} [Topology 11, 307-315 (1972; Zbl 0237.58015)] constructs examples of flat manifolds admitting (or not) Anosov diffeomorphisms. In particular he proves: 1. In each dimension greater than or equal to 6, there exists a flat manifold with first Betti number zero and holonomy \((\mathbb{Z}_2\oplus \mathbb{Z}_2)\), admitting Anosov diffeomorphisms. 2. Fix an integer \(k>1\). In dimension \(n\), there exists a flat manifold \(M\) with first Betti number \(k\) and admitting Anosov diffeomorphisms if and only if \(n=k\) or \(n\geq k+2\).