Convex and Concave Decompositions of Affine 3-Manifolds

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 $${\mathbf {Aff}}({{\mathbb {R}}}^3)$$. 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 connected open set in $${{\mathbb {R}}}^{3}$$, then M is irreducible or is an affine Hopf manifold.