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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
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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 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.

Readership

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

Table of Contents

  • Introduction
  • Acknowledgement
  • Preliminary
  • \((n-1)\)-convexity: previous results
  • Radiant vector fields, generalized affine suspensions, and the radial completeness
  • 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
  • The radiant tetrahedron case
  • The radiant trihedron case
  • Obtaining concave-cone affine manifolds
  • Concave-cone radiant affine \(3\)-manifolds and radiant concave affine \(3\)-manifolds
  • The nonexistence of pseudo-crescent-cones
  • Appendix A. Dipping intersections
  • Appendix B. Sequences of \(n\)-balls
  • Appendix C. Radiant affine \(3\)-manifolds with boundary, and certain radiant affine \(3\)-manifolds
  • Bibliography
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