The Decomposition and Classification of Radiant Affine 3-Manifolds (Memoirs of the American Mathematical Society)

The Decomposition and Classification of Radiant Affine 3-Manifolds (Memoirs of the American Mathematical Society)

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Description

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 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 cross-section, confirming a conjecture of Carriere, 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.

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.

Product Details

  • ISBN13: 9780821827048
  • Format: Paperback
  • ID: 9780821827048
  • ISBN10: 0821827049

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