The universal cover of an affine three-manifold with holonomy of shrinkable dimension $\leq 2$

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dc.creatorChoi, Suhyoung
dc.date1997-06-23
dc.date1999-12-21
dc.date.accessioned2026-07-07T08:59:13Z
dc.date.available2026-07-07T08:59:13Z
dc.descriptionAn 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 discompacité 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.
dc.description67 pages, 27 figures, some revisions. To appear in the International Journal Of Mathematics
dc.identifierhttps://arxiv.org/abs/dg-ga/9706011
dc.identifierhttp://arxiv.org/abs/dg-ga/9706011
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/147633
dc.subjectDifferential Geometry
dc.subject57M50 (Primary) 53A20, 53C15(Secondary)
dc.titleThe universal cover of an affine three-manifold with holonomy of shrinkable dimension $\leq 2$
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