The universal cover of an affine three-manifold with holonomy of shrinkable dimension <= two

Abstract

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 discompacite in French) less than or equal to two is diffeomorphic to R-3. Hence, M is irreducible. This follows from two results: (i) a simply connected affine 3-manifold which is 2-convex is diffeomorphic to R-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 R-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 R-3 A Cake cell has an affine structure as shown by Gromov. Such a cell must have a concave point at the boundary.