Classification of affine equivalence classes of $\mathbb{Z}_p$-manifolds using Bieberbach groups

Abstract

This work investigates the classification of $\mathbb{Z}_p$-manifolds, compact (pathconnected) Riemannian manifolds whose holonomy group is isomorphic to $\mathbb{Z}_p$, up to affine equivalence. It uses the foundational results of Bieberbach groups and cohomological methods to achieve two primary objectives: classifying affine equivalence classes of $\mathbb{Z}_p$-manifolds and analyzing the case where non-homotopic $\mathbb{Z}_p$-manifolds become affinely equivalent when taking the product by $S^1$. This work also provides a way to find pairs of such non-homotopic $\mathbb{Z}_p$-manifolds that become isomorphic after taking Cartesian product by $S^1$.

Notation: In this work, $\mathbb{Z}_p$ refers to $\mathbb{Z} / p \mathbb{Z}$, where $p \in \mathbb{Z}, p \neq 0$.