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**The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold.**
*(English)*
Zbl 1277.57016

The paper under review deals with the (generalized) Smale conjecture for Seifert fibered 3-manifolds with basis a hyperbolic 2-orbifold. This is a significant contribution to the generalized Smale conjecture for 3-manifolds, as well as to the study of the topology of the space of Seifert fibrations of a manifold.

The Smale conjecture (proved by Jean Cerf and Allen Hatcher) asserts that, for the 3-sphere equipped with the round metric, the inclusion of the isometry group into the group of diffeomorphisms is a homotopy equivalence. The generalization fails for some Haken 3-manifolds, because there are connected components of the group of diffeomorphisms that contain no isometry (even if one takes the metric on the 3-manifold with maximal symmetry). One may discuss however the inclusion of the identity components of both, and ask whether the inclusion of the component of the identity of the group of isometries is homotopically equivalent to the component of the group of diffeomorphisms. The homotopy equivalence of identity components was known to hold in many cases (Haken, hyperbolic, Euclidean).

The paper under review proves the identity component version of the Smale conjecture for Seifert manifolds with hyperbolic basis. This completes it for all aspherical 3-manifolds except for non-Haken infranil-manifolds (non-Haken with nil geometry).

Combining this result with [S. Hong et al., Diffeomorphisms of elliptic 3-manifolds. Lecture Notes in Mathematics 2055. Berlin: Springer (2012; Zbl 1262.57001)], the authors also prove that, for a compact orientable aspherical 3-manifold different from a small infranil-manifold, each component of the space of Seifert fibrations is contractible.

For the proof, the authors use that the group of diffeomorphisms has the homotopy type of a CW-complex and they compute its homotopy groups. According to the authors, their proof incorporates many of the ideas of Gabai’s proof of the Smale conjecture for hyperbolic 3-manifolds in [D. Gabai, J. Differ. Geom. 58, No. 1, 113–149 (2001; Zbl 1030.57026)]. Gabai’s proof relies on his rigidity theorem for hyperbolic 3-manifolds [J. Am. Math. Soc. 10, No. 1, 37–74 (1997; Zbl 0870.57014)], instead the authors use Scott’s rigidity of Seifert fibered spaces, cf. T. Soma [Trans. Am. Math. Soc. 358, No. 9, 4057–4070 (2006; Zbl 1096.57014)].

The Smale conjecture (proved by Jean Cerf and Allen Hatcher) asserts that, for the 3-sphere equipped with the round metric, the inclusion of the isometry group into the group of diffeomorphisms is a homotopy equivalence. The generalization fails for some Haken 3-manifolds, because there are connected components of the group of diffeomorphisms that contain no isometry (even if one takes the metric on the 3-manifold with maximal symmetry). One may discuss however the inclusion of the identity components of both, and ask whether the inclusion of the component of the identity of the group of isometries is homotopically equivalent to the component of the group of diffeomorphisms. The homotopy equivalence of identity components was known to hold in many cases (Haken, hyperbolic, Euclidean).

The paper under review proves the identity component version of the Smale conjecture for Seifert manifolds with hyperbolic basis. This completes it for all aspherical 3-manifolds except for non-Haken infranil-manifolds (non-Haken with nil geometry).

Combining this result with [S. Hong et al., Diffeomorphisms of elliptic 3-manifolds. Lecture Notes in Mathematics 2055. Berlin: Springer (2012; Zbl 1262.57001)], the authors also prove that, for a compact orientable aspherical 3-manifold different from a small infranil-manifold, each component of the space of Seifert fibrations is contractible.

For the proof, the authors use that the group of diffeomorphisms has the homotopy type of a CW-complex and they compute its homotopy groups. According to the authors, their proof incorporates many of the ideas of Gabai’s proof of the Smale conjecture for hyperbolic 3-manifolds in [D. Gabai, J. Differ. Geom. 58, No. 1, 113–149 (2001; Zbl 1030.57026)]. Gabai’s proof relies on his rigidity theorem for hyperbolic 3-manifolds [J. Am. Math. Soc. 10, No. 1, 37–74 (1997; Zbl 0870.57014)], instead the authors use Scott’s rigidity of Seifert fibered spaces, cf. T. Soma [Trans. Am. Math. Soc. 358, No. 9, 4057–4070 (2006; Zbl 1096.57014)].

Reviewer: Joan Porti (Bellaterra)