Memoirs of the American Mathematical Society 2001; 122 pp; softcover Volume: 154 ISBN10: 0821827049 ISBN13: 9780821827048 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 torsionfree 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 concavecone 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 crosssection, 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 crosssections, 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
 \((n1)\)convexity: previous results
 Radiant vector fields, generalized affine suspensions, and the radial completeness
 Threedimensional 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 concavecone affine manifolds
 Concavecone radiant affine \(3\)manifolds and radiant concave affine \(3\)manifolds
 The nonexistence of pseudocrescentcones
 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
