### 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 language | English |
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Title of host publication | Diffeomorphisms of Elliptic 3-Manifolds |

Publisher | Springer Verlag |

Pages | 53-83 |

Number of pages | 31 |

ISBN (Print) | 9783642315633 |

DOIs | |

Publication status | Published - 2012 Jan 1 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 2055 |

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

*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