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The Decomposition and Classification of Radiant Affine 3-Manifolds
Suhyoung Choi, Seoul National University, Korea
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Memoirs of the American Mathematical Society
2001; 122 pp; softcover
Volume: 154
ISBN-10: 0-8218-2704-9
ISBN-13: 978-0-8218-2704-8
List Price: US$54 Individual Members: US$32.40
Institutional Members: US\$43.20
Order Code: MEMO/154/730

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.

Graduate students and research mathematicians interested in manifolds and cell complexes, and differential geometry.

• Introduction
• Acknowledgement
• Preliminary
• $$(n-1)$$-convexity: previous results
• Three-dimensional radiant affine manifolds and concave affine manifolds
• The decomposition along totally geodesic surfaces
• $$2$$-convex radiant affine manifolds
• The claim and the rooms
• Concave-cone radiant affine $$3$$-manifolds and radiant concave affine $$3$$-manifolds
• Appendix B. Sequences of $$n$$-balls
• Appendix C. Radiant affine $$3$$-manifolds with boundary, and certain radiant affine $$3$$-manifolds