Flat manifolds with holonomy group Z_2^k of diagonal type

TL;DR

This paper explores the relationship between two families of flat manifolds with holonomy group (Z_2)^k, showing their intersection is non-empty and analyzing conditions for certain spin structures.

Contribution

It establishes the non-empty intersection between real Bott manifolds and generalized Hantzsche-Wendt manifolds and provides conditions for the existence of spin structures.

Findings

The intersection of GHW and RBM families is non-empty.

Conditions for the non-existence of Spin and Spin^C structures are given.

Insights into the structure of flat manifolds with holonomy group (Z_2)^k.

Abstract

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