Non-orientable 3-manifolds of surface-complexity one

TL;DR

This paper classifies all non-orientable, face-identified cube 3-manifolds with surface-complexity one, revealing that they are exactly the four flat non-orientable 3-manifolds.

Contribution

It provides a complete classification of non-orientable, face-identified cube 3-manifolds with surface-complexity one, identifying them as the four flat non-orientable manifolds.

Findings

All such manifolds are the four flat non-orientable 3-manifolds.

The classification is achieved through face identification of a cube.

Surface-complexity one characterizes these specific manifolds.

Abstract

We classify all closed non-orientable $\mathbb{P}^2$-irreducible 3-manifolds obtained by identifying the faces of a cube. These turn out to be the closed non-orientable $\mathbb{P}^2$-irreducible 3-manifolds with surface-complexity one. We show that they are the four flat ones.