  An affine manifold is a manifold with an affine structure, i.e. a
torsion-free flat affine connection. We show that the universal cover of a
closed affine 3-manifold $M$ with holonomy group of shrinkable dimension (or
discompacit\'e in French) less than or equal to two is diffeomorphic to
$\bR^3$. Hence, $M$ is irreducible. This follows from two results: (i) a simply
connected affine 3-manifold which is 2-convex is diffeomorphic to $\bR^3$,
whose proof using the Morse theory takes most of this paper; and (ii) a closed
affine manifold of holonomy of shrinkable dimension less or equal to $d$ is
$d$-convex. To prove (i), we show that 2-convexity is a geometric form of
topological incompressibility of level sets. As a consequence, we show that the
universal cover of a closed affine three-manifold with parallel volume form is
diffeomorphic to $\bR^3$, a part of the weak Markus conjecture. As
applications, we show that the universal cover of a hyperbolic 3-manifold with
cone-type singularity of arbitrarily assigned cone-angles along a link removed
with the singular locus is diffeomorphic to $\bR^3$. A fake cell has an affine
structure as shown by Gromov. Such a cell must have a concave point at the
boundary.
