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**Diffeomorphisms of elliptic 3-manifolds.**
*(English)*
Zbl 1262.57001

Lecture Notes in Mathematics 2055. Berlin: Springer (ISBN 978-3-642-31563-3/pbk; 978-3-642-31564-0/ebook). x, 155 p. (2012).

The aim of this book is to present detailed proofs of the generalized Smale conjecture for two large classes of elliptic manifolds. Recall that a compact 3-manifold is defined to be elliptic if it admits a Riemannian metric of constant positive curvature. Since the Poincaré conjecture has now been proved, a compact 3-manifold is known to be elliptic if and only if its fundamental group is finite. The generalized Smale conjecture is the assertion that for an elliptic manifold \(M\) the inclusion of the group \(Isom(M)\) of isometries of \(M\) into its group \(Diff(M)\) of diffeomorphisms is a homotopy equivalence. In the case \(M = S^3\), one has the original Smale conjecture that was proved by Cerf and Hatcher. In this book the generalized Smale conjecture is proved in the cases either (1) when \(M\) contains a geometrically incompressible Klein bottle or (2) when \(M\) is a lens space \(L(m,q)\) with \(m \geq 3\). Certain subcases of (1) were previously obtained by Ivanov, but the authors extended his result to the full case. The authors remark that the conjecture is still open when \(M\) is real projective 3-space or is a tetrahedral, octahedral, or icosahedral 3-manifold.

The proof of the generalized Smale conjecture utilizes various facts about the group of the diffeomorphisms of a manifold and the space of embeddings of a submanifold. For completeness, this book provides proofs of these facts in the form that they are needed. So here one will find, in particular, proofs that these spaces are Fréchet manifolds and have the homotopy type of a CW complex in the general case of manifolds with boundary. Moreover, if \(f:E \rightarrow B\) is a fibration or singular fibration, the authors study the relations among the group \(Diff(E)\), the group \(Diff_f(E)\) of fiber preserving diffeomorphisms, and the space \(Diff(E)/Diff_f(E)\) of equivalent fiberings, most notably in the case when \(f\) is a Seifert fibering. In this context, they give versions of the restriction fibration theorems of Cerf and Palais that are needed in the proof of the Smale conjecture.

The proof of the generalized Smale conjecture utilizes various facts about the group of the diffeomorphisms of a manifold and the space of embeddings of a submanifold. For completeness, this book provides proofs of these facts in the form that they are needed. So here one will find, in particular, proofs that these spaces are Fréchet manifolds and have the homotopy type of a CW complex in the general case of manifolds with boundary. Moreover, if \(f:E \rightarrow B\) is a fibration or singular fibration, the authors study the relations among the group \(Diff(E)\), the group \(Diff_f(E)\) of fiber preserving diffeomorphisms, and the space \(Diff(E)/Diff_f(E)\) of equivalent fiberings, most notably in the case when \(f\) is a Seifert fibering. In this context, they give versions of the restriction fibration theorems of Cerf and Palais that are needed in the proof of the Smale conjecture.

Reviewer: James Hebda (St. Louis)

### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57M99 | General low-dimensional topology |

57S10 | Compact groups of homeomorphisms |

58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |

58D29 | Moduli problems for topological structures |

57R50 | Differential topological aspects of diffeomorphisms |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |