Homology, Homotopy and Applications

Volume 21 (2019)

Number 2

Spin structures of flat manifolds of diagonal type

Pages: 333 – 344

DOI: https://dx.doi.org/10.4310/HHA.2019.v21.n2.a18


Rafał Lutowski (Institute of Mathematics, University of Gdańsk, Poland)

Nansen Petrosyan (Mathematical Sciences, University of Southampton, United Kingdom)

Jerzy Popko (Institute of Mathematics, University of Gdańsk, Poland)

Andrzej Szczepański (Institute of Mathematics, University of Gdańsk, Poland)


We give a novel and purely combinatorial description of Stiefel–Whitney classes of closed flat manifolds with diagonal holonomy representation. Using this description, for each integer $d$ at least two, we construct non-spin closed oriented flat manifolds with holonomy group $\mathbb{Z}^d_2$ with the property that all of their finite proper covers have a spin structure. Moreover, all such covers have trivial Stiefel–Whitney classes. In contrast to the case of real Bott manifolds, this shows that for a general closed flat manifold the existence of a spin structure may not be detected by its finite proper covers.


flat manifold, crystallographic group, spin structure

2010 Mathematics Subject Classification

20H15, 53C27

The first and fourth authors were supported by the Polish National Science Center grant 2013/09/B/ST1/04125. The second author was supported by the EPSRC First Grant EP/N033787/1.

Received 26 January 2018

Received revised 4 October 2018

Published 3 April 2019