Elliptic Three-Manifolds and the Smale Conjecture

After a discussion of the (Generalized) Smale Conjecture, the main results of the monograph are summarized. The extent to which the Smale Conjecture extends to larger classes of three-manifolds—usually in a limited form called the Weak Smale Conjecture, if at all—is detailed. The chapter closes with a brief discussion of why Perelman’s methods appear not to give progress on the Smale Conjecture. As noted in the Preface, theSmale Conjecture is the assertion that the inclusion is a homotopy equivalence whenever M is an elliptic three-manifold, that is, a three-manifold with a Riemannian metric of constant positive curvature (which may be assumed to be 1). TheGeometrization Conjecture, now proven byPerelman, shows that all closed three-manifolds with finite fundamental group are elliptic.In this chapter, we will first review elliptic three-manifolds and their isometry groups. In the second section, we will state our main results on the Smale Conjecture, and provide some historical context. In the final two sections, we discuss isometries of nonelliptic three-manifolds, and address the possibility of applying Perelman’s methods to the Smale Conjecture.