Convex and concave decompositions of affine $3$-manifolds

TL;DR

This paper classifies closed affine 3-manifolds, showing they are either affine Hopf manifolds or decompose into canonical submanifolds, with implications for their primeness and irreducibility.

Contribution

It introduces a canonical decomposition of affine 3-manifolds into simpler components, extending understanding of their structure and classification.

Findings

Connected closed affine 3-manifolds are either affine Hopf or decompose into canonical submanifolds.

If no toral π-submanifold exists, the manifold is prime.

Manifolds covered by a domain in Euclidean space are either irreducible or affine Hopf.

Abstract

A (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space $\mathbb{R}^3$ with transition maps in the affine transformation group $\mathrm{Aff}(\mathbb{R}^3)$. We will show that a connected closed affine $3$-manifold is either an affine Hopf $3$-manifold or decomposes canonically to concave affine submanifolds with incompressible boundary, toral $\pi$-submanifolds and $2$-convex affine manifolds, each of which is an irreducible $3$-manifold. It follows that if there is no toral $\pi$-submanifold, then $M$ is prime. Finally, we prove that if a closed affine manifold is covered by a domain in $\mathbb{R}^{n}$, then $M$ is irreducible or is an affine Hopf manifold.