Hantzsche-Wendt manifolds (joint talk on the conference and the Seminar of the Laboratory of Algebraic Geometry at HSE)

There are flat Riemannian manifolds of odd dimension n with holonomy group $(\mathbb{Z}_2)^{n-1}$. From Bieberbach theorems its fundamental group G is torsion free and defines a short exact sequence $0 \to \mathbb{Z}^n \to G \to (\mathbb{Z}_2)^{n-1}\to 0$, where $\mathbb{Z}^n$ is a maximal abelian subgroup in G. This class of manifolds (groups) has many very interesting properties:

• they are rational homology spheres
• they are homology rigid i.e $M$ is diffeomorphic to $M'$ if and only if cohomology rings $H^{*}(M,F_2)$ and $H^{*}(M',F_2)$ are isomorphic
• they have no Spin structure