Spin structures of flat manifolds of diagonal type

TL;DR

This paper constructs specific flat manifolds of diagonal type with holonomy groups of the form Z_2^d, which are non-spin themselves but have all finite proper covers that are spin and have trivial Stiefel-Whitney classes.

Contribution

It introduces a new class of flat manifolds with particular spin and cover properties, expanding understanding of spin structures in geometric topology.

Findings

Existence of non-spin flat manifolds with all finite proper covers being spin.

All such covers have trivial Stiefel-Whitney classes.

Construction method for these manifolds.

Abstract

For each integer $d$ at least two, we construct non-spin closed oriented flat manifolds with holonomy group $\mathbb Z_2^d$ and with the property that all of their finite proper covers have a spin structure. Moreover, all such covers have trivial Stiefel-Whitney classes.