The universal cover of an affine three-manifold with holonomy of shrinkable dimension \(\leq\) two. (English) Zbl 1110.57301
Summary: 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 discompacity in French) less than or equal to two is diffeomorphic to \(\mathbb R^3\). Hence, \(M\) is irreducible. This follows from two results: (i) a simply connected affine 3-manifold which is 2-convex is diffeomorphic to \(\mathbb R^3\), the proof of which using 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 \(\mathbb R^3\), 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 \(\mathbb R^3\). A fake cell has an affine structure as shown by Gromov. Such a cell must have a concave point at the boundary.
MSC:
| 57M50 | General geometric structures on low-dimensional manifolds |
| 53C29 | Issues of holonomy in differential geometry |
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
Keywords:
Affine manifold; Morse theory; aspherical three-manifold; affine structure; singular hyperbolic three-manifold; fake 3-cellReferences:
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