Ricci flow and diffeomorphism groups of 3-manifolds

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We complete the proof of the Generalized Smale Conjecture, apart from the case of RP³, and give a new proof of Gabai's theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than S³ and RP³ and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold X, the inclusion Isom (X,g)→ Diff(X) is a homotopy equivalence for any Riemannian metric g of constant sectional curvature.