Geodesic
Spin structures of flat manifolds of diagonal type
Homology, homotopy, and applications, Tome 21 (2019) no. 2, pp. 333-344.

Voir la notice de l'article provenant de la source International Press of Boston

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.
DOI : 10.4310/HHA.2019.v21.n2.a18
Classification : 20H15, 53C27
Keywords: flat manifold, crystallographic group, spin structure 
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     title = {Spin structures of flat manifolds of diagonal type},
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     pages = {333--344},
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Rafał Lutowski; Nansen Petrosyan; Jerzy Popko; Andrzej Szczepański. Spin structures of flat manifolds of diagonal type. Homology, homotopy, and applications, Tome 21 (2019) no. 2, pp. 333-344. doi : 10.4310/HHA.2019.v21.n2.a18. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2019.v21.n2.a18/

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