Flat manifolds with holonomy group Z_2^k of diagonal type

Abstract

We consider relations between two families of flat manifolds with holonomy group (Z_2)^k of diagonal type. The family $\mathcal{R}\mathcal{BM}$ of real Bott manifolds and the family $\mathcal{GH}\mathcal{W}$ of generalized Hantzsche-Wendt manifolds. In particular, we prove that the intersection $\mathcal{GH}\mathcal{W}\cap \mathcal{R}\mathcal{BM}$ is not empty. We also consider some class of real Bott manifolds without $\mathrm{Spin}$ and $\operatorname{Spin}^{\C}$ structure. There are given conditions for the (non)existence of such structures.

12 pages