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Mathematics - Geometry & Topology | Diffeomorphisms of Elliptic 3-Manifolds

Diffeomorphisms of Elliptic 3-Manifolds

Series: Lecture Notes in Mathematics, Vol. 2055

Hong, S., Kalliongis, J., McCullough, D., Rubinstein, J.H.

2012, X, 155 p. 22 illus.

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This work concerns the diffeomorphism groups of 3-manifolds, in particular of elliptic 3-manifolds. These are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, now known to be exactly the closed 3-manifolds that have a finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to its diffeomorphism group is a homotopy equivalence. The original Smale Conjecture, for the 3-sphere, was proven by J. Cerf and A. Hatcher, and N. Ivanov proved the generalized conjecture for many of the elliptic 3-manifolds that contain a geometrically incompressible Klein bottle.

The main results establish the Smale Conjecture for all elliptic 3-manifolds containing geometrically incompressible Klein bottles, and for all lens spaces L(m,q) with m at least 3. Additional results imply that for a Haken Seifert-fibered 3 manifold V, the space of Seifert fiberings has contractible components, and apart from a small list of known exceptions, is contractible. Considerable foundational and background material on diffeomorphism groups is included.

Content Level » Research

Keywords » 3-manifold - 57M99, 57S10, 58D05, 58D29 - Frechet - Smale Conjecture - elliptic

Related subjects » Geometry & Topology

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