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 R3 with transition maps in the affine transformation group Aff(R3). 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 π-submanifolds and 2-convex affine manifolds, each of which is an irreducible 3-manifold. It follows that if there is no toral π-submanifold, then M is prime. Finally, we prove that if a closed affine manifold is covered by a connected open set in R3, then M is irreducible or is an affine Hopf manifold.