Convex and concave decompositions of affine $3$-manifolds

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.