Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
The Decomposition and Classification of Radiant Affine 3-Manifolds
 
Suhyoung Choi Seoul National University, Seoul, Korea
The Decomposition and Classification of Radiant Affine 3-Manifolds
eBook ISBN:  978-1-4704-0323-2
Product Code:  MEMO/154/730.E
List Price: $57.00
MAA Member Price: $51.30
AMS Member Price: $34.20
The Decomposition and Classification of Radiant Affine 3-Manifolds
Click above image for expanded view
The Decomposition and Classification of Radiant Affine 3-Manifolds
Suhyoung Choi Seoul National University, Seoul, Korea
eBook ISBN:  978-1-4704-0323-2
Product Code:  MEMO/154/730.E
List Price: $57.00
MAA Member Price: $51.30
AMS Member Price: $34.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1542001; 122 pp
    MSC: Primary 57; Secondary 53

    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
     
     
    • Chapters
    • 0. Introduction
    • 1. Preliminary
    • 2. ($n$ - 1)-convexity: previous results
    • 3. Radiant vector fields, generalized affine suspensions, and the radial completeness
    • 4. Three-dimensional radiant affine manifolds and concave affine manifolds
    • 5. The decomposition along totally geodesic surfaces
    • 6. 2-convex radiant affine manifolds
    • 7. The claim and the rooms
    • 8. The radiant tetrahedron case
    • 9. The radiant trihedron case
    • 10. Obtaining concave-cone affine manifolds
    • 11. Concave-cone radiant affine 3-manifolds and radiant concave affine 3-manifolds
    • 12. The nonexistence of pseudo-crescent-cones
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1542001; 122 pp
MSC: Primary 57; Secondary 53

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.

  • Chapters
  • 0. Introduction
  • 1. Preliminary
  • 2. ($n$ - 1)-convexity: previous results
  • 3. Radiant vector fields, generalized affine suspensions, and the radial completeness
  • 4. Three-dimensional radiant affine manifolds and concave affine manifolds
  • 5. The decomposition along totally geodesic surfaces
  • 6. 2-convex radiant affine manifolds
  • 7. The claim and the rooms
  • 8. The radiant tetrahedron case
  • 9. The radiant trihedron case
  • 10. Obtaining concave-cone affine manifolds
  • 11. Concave-cone radiant affine 3-manifolds and radiant concave affine 3-manifolds
  • 12. The nonexistence of pseudo-crescent-cones
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.