Flat manifolds with holonomy group $\mathbb{Z}_{2}^{k}$ of diagonal type

Abstract

We consider relations between two families of flat manifolds with holonomy group $\mathbb{Z}_{2}^{k}$ of diagonal type: the family $\mathcal{RBM}$ of real Bott manifolds and the family $\mathcal{GHW}$ of generalized Hantzsche--Wendt manifolds. In particular, we prove that the intersection $\mathcal{GHW}\cap \mathcal{RBM}$ is not empty. Moreover, we consider some class of real Bott manifolds without $\operatorname{Spin}$ and $\operatorname{Spin}^{\mathbb{C}}$ structure. There are given conditions (Theorem 1) for the existence of such structures. As an application a list of all 5-dimensional oriented real Bott manifolds without $\operatorname{Spin}$ structure is given.