Diffeomorphism groups of prime 3-manifolds

Abstract

Abstract Let X be a compact orientable non-Haken 3-manifold modeled on the Thurston geometry Nil {\operatorname{Nil}} . We show that the diffeomorphism group Diff ⁡ ( X ) {\operatorname{Diff}(X)} deformation retracts to the isometry group Isom ⁡ ( X ) {\operatorname{Isom}(X)} . Combining this with earlier work by many authors, this completes the determination the homotopy type of Diff ⁡ ( X ) {\operatorname{Diff}(X)} for any compact, orientable, prime 3-manifold X.