The Smale Conjecture for lens spaces

Abstract

The original Smale Conjecture asserted that the inclusion of the group O(4) of isometries of the round 3-sphere S into the full diffeomorphism group Diff(S) is a homotopy equivalence. The (Generalized) Smale Conjecture asserts that the inclusion of Isom(M) into Diff(M) is a homotopy equivalence whenever M is an elliptic 3-manifold, that is, a closed Riemannian 3-manifold of constant positive curvature. We prove the Smale Conjecture for all lens spaces L(m,q), where m is at least 3.