Lens Spaces

The Smale Conjecture is proven for all lens spaces other than L(2, 1), the real projective three-space. The method utilizes the Rubinstein–Scharlemann graphic, in this case comparing a sweepout obtained from a genus-1 Heegaard splitting of a lens space with its images under a parameterized family of diffeomorphisms. Examples show that the family may need to undergo a perturbation by a small homotopy, after which the graphic yields at each parameter a loop in the intersection of two Heegaard tori that is essential in each of them. Using methods of Hatcher for working with parameterized families of diffeomorphisms, these essential intersections eventually lead to a deformation of the family to preserve a fixed Seifert fibration of the lens space, and the Conjecture is deduced from this.Recall that we always use the term lens space to mean a three-dimensional lens space L(m, q) with. In addition, we always select q so that. In this chapter, we will prove Theorem 1.3, the Smale Conjecture for Lens Spaces. The argument is regrettably quite lengthy. It uses a lot of combinatorial topology, but draws as well on some mathematics unfamiliar to many low-dimensional topologists. We have already seen some of that material in earlier chapters, but we will also have to use the Rubinstein–Scharlemann method, reviewed in Sect. 5.6, and some results from singularity theory, presented in Sect. 5.8.The next section is a comprehensive outline of the entire proof. We hope that it will motivate the various technical complications that ensue.