Flat manifolds with prescribed first Betti number admitting Anosov diffeomorphisms

We show that from dimension six onwards (but not in lower dimensions), ther e are in each dimension flat manifolds with first Betti number equal to zer o admitting Anosov diffeomorphisms. On the other hand, it is known that no flat manifolds with first Betti number equal to one support Anosov diffeomo rphisms. For each integer k > 1 however, we prove that there is an n-dimens ional flat manifold M with first Betti number equal to k carrying an Anosov diffeomorphism if and only if M is a k-torus or n is greater than or equal to k + 2.