Elliptic Three-Manifolds Containing One-Sided Klein Bottles

Sungbok Hong, John Kalliongis, Darryl McCullough, J. Hyam Rubinstein

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Extending work of N. Ivanov, the Smale Conjecture is proven for all elliptic three-manifolds containing one-sided Klein bottles, other than the lens space L(4, 1). The technique takes a parameterized family of diffeomorphisms and uses its restriction to embeddings of the Klein bottles to deform the diffeomorphisms to preserve a Seifert fibration of the manifolds. The Conjecture is deduced from this. Another key element of the proof is a collection of techniques for working with parameterized families developed by A. Hatcher.In this chapter, we will prove Theorem 1.3. Section 4.1 gives a construction of the elliptic three-manifolds that contain a one-sided geometrically incompressible Klein bottle; they are described as a family of manifolds M(m, n) that depend on two integer parameters. Section 4.2 is a section-by-section outline of the entire proof, which constitutes the remaining sections of the chapter.

Original languageEnglish
Title of host publicationDiffeomorphisms of Elliptic 3-Manifolds
PublisherSpringer Verlag
Pages53-83
Number of pages31
ISBN (Print)9783642315633
DOIs
Publication statusPublished - 2012 Jan 1

Publication series

NameLecture Notes in Mathematics
Volume2055
ISSN (Print)0075-8434
ISSN (Electronic)1617-9692

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Keywords

  • Cone Point
  • Free Abelian Group
  • Inverse Image
  • Isometry Group
  • Parameterized Family

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Hong, S., Kalliongis, J., McCullough, D., & Rubinstein, J. H. (2012). Elliptic Three-Manifolds Containing One-Sided Klein Bottles. In Diffeomorphisms of Elliptic 3-Manifolds (pp. 53-83). (Lecture Notes in Mathematics; Vol. 2055). Springer Verlag. https://doi.org/10.1007/978-3-642-31564-0_4