Diffeomorphism groups of prime 3-manifolds

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

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