Elliptic Three-Manifolds Containing One-Sided Klein Bottles

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.