# The Decomposition and Classification of Radiant Affine 3-Manifolds

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*Suhyoung Choi*

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 Carrière, 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.

#### Table of Contents

# Table of Contents

## The Decomposition and Classification of Radiant Affine 3-Manifolds

- Contents vii8 free
- Chapter 0. Introduction 110 free
- Acknowledgement 615 free

- Chapter 1. Preliminary 817
- Chapter 2. (n – l)-convexity: previous results 1726
- Chapter 3. Radiant vector fields, generalized affine suspensions, and the radial completeness 2029
- Chapter 4. Three-dimensional radiant affine manifolds and concave affine manifolds 3039
- Chapter 5. The decomposition along totally geodesic surfaces 3443
- Chapter 6. 2-convex radiant affine manifolds 3746
- Chapter 7. The claim and the rooms 4554
- Chapter 8. The radiant tetrahedron case 4958
- Chapter 9. The radiant trihedron case 5564
- Chapter 10. Obtaining concave-cone affine manifolds 6776
- Chapter 11. Concave-cone radiant affine 3-manifolds and radiant concave affine 3-manifolds 7584
- Chapter 12. The nonexistence of pseudo-crescent-cones 8493
- Appendix A. Dipping intersections 94103
- Appendix B. Sequences of n-balls 96105
- Appendix C. Radiant affine 3-manifolds with boundary, and certain radiant affine 3-manifolds 98107
- Bibliography 121130