Classification of spin structures on 4-dimensional almost-flat manifolds

TL;DR

This paper classifies spin structures on four-dimensional almost-flat manifolds, revealing that most are parallelizable with only a few non-spin cases, thus extending understanding of their geometric properties.

Contribution

It provides a complete classification of spin structures on all non-flat four-dimensional almost-flat manifolds, based on their fundamental group representations.

Findings

15 non-spin manifolds among 127 families

Most four-dimensional almost-flat manifolds are parallelizable

Classification based on canonical orthogonal representation

Abstract

Almost-flat manifolds were defined by Gromov as a natural generalisation of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost-flat manifold is determined by the canonical orthogonal representation of its fundamental group. Utilising this, we classify the spin structures on all four-dimensional almost-flat manifolds that are not flat. Out of 127 orientable families, there are exactly 15 that are non-spin, the rest are in fact parallelizable.