# Integral affine 3-manifolds

## TL;DR

This paper classifies all complete integral affine structures on compact 3-manifolds, including tori and nilmanifolds, providing a comprehensive understanding of their geometric configurations.

## Contribution

It offers a complete classification of integral affine structures on compact three-dimensional manifolds, including explicit lists for tori and nilmanifolds, up to finite coverings.

## Key findings

- Complete classification of integral affine structures on compact 3-manifolds
- Explicit list of structures on 3-torus and nilmanifolds
- Descriptions up to finite-sheeted coverings

## Abstract

Affine manifolds are called integral if there is an atlas such that all transition maps are affine transformations with integer matrices of linear parts. In this paper we describe all complete integral affine structures on compact three-dimensional manifolds up to a finite-sheeted covering. Also a complete list of integral affine structures on the three-dimensional torus and compact three-dimensional nilmanifolds was obtained.