The Smale conjecture for hyperbolic 3-manifolds: \(\text{Isom}(M^3)\simeq\text{Diff}(M^3)\)
The main result of the paper states that, for a closed hyperbolic 3-manifold \(M\), the inclusion of the (finite) isometry group \(\text{Isom}(M)\) into the diffeomorphism group \(\text{Diff}(M)\) is a homotopy equivalence. For Haken 3-manifolds, this follows from results of \textit{A. Hatcher} [Topology 15, 343-347 (1976; Zbl 0335.57004)] and Ivanov (the original Smale conjecture, proved by Hatcher, is for the case of the 3-sphere). By previous work of \textit{D. Gabai} [J. Am. Math. Soc. 10, 37-74 (1997; Zbl 0870.57014)] and \textit{D. Gabai, G. R. Meyerhoff} and \textit{N. Thurston} [Ann. Math. (2) 157, 335-431 (2003)], the inclusion \(\text{Isom}(M)\to \text{Diff}(M)\) induces a bijection of \(\pi_0\text{Isom}(M)\) and \(\pi_0\text{Diff}(M)\). The main result is then equivalent to the fact that the path component \(\text{Diff}_0(M)\) in \(\text{Diff}(M)\) of the identity map of \(M\) is contractible. For the proof, the contractibility of the space of Riemannian metrics on \(M\) is used, in conjunction with the insulator theory of the above-mentioned paper of Gabai, to reduce to the Haken case. It follows also that the space of hyperbolic metrics on a complete hyperbolic 3-manifold of finite volume is contractible.
scientific article; zbMATH DE number 1782691