The Decomposition and Classification of Radiant Affine 3-Manifolds

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An affine manifold is a manifold with torsion-free flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with a holonomy group consisting of affine transformations fixing a common fixed point. We decompose a closed radiant affine 3-manifold into radiant 2-convex affine manifolds and radiant concave affine 3-manifolds along mutually disjoint totally geodesic tori or Klein bottles using the convex and concave decomposition of real projective n-manifolds developed earlier. Then we decompose a 2-convex radiant affine manifold into convex radiant affine manifolds and concave-cone affine manifolds. To do this, we will obtain certain nice geometric objects in the Kuiper completion of a holonomy cover. The equivariance and local finiteness property of the collection of such objects will show that their union covers a compact submanifold of codimension zero, the complement of which is convex. Finally, using the results of Barbot, we will show that a closed radiant affine 3-manifold admits a total cross-section, confirming a conjecture of Carriere, and hence every closed radiant affine 3-manifold is homeomorphic to a Seifert fibered space with trivial Euler number, or a virtual bundle over a circle with fiber homeomorphic to a Euler characteristic zero surface. In Appendix C, Thierry Barbot and the author show the nonexistence of certain radiant affine 3-manifolds and that compact radiant affine 3-manifolds with nonempty totally geodesic boundary admit total cross-sections, which are key results for the main part of the paper.
Publisher
AMER MATHEMATICAL SOC
Issue Date
2001-11
Language
English
Article Type
Article
Keywords

REAL PROJECTIVE-STRUCTURES; CONVEX DECOMPOSITIONS; COMPACT SURFACES; MANIFOLDS

Citation

MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY, v.154, no.730, pp.1 - 1

ISSN
0065-9266
URI
http://hdl.handle.net/10203/83464
Appears in Collection
MA-Journal Papers(저널논문)
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