Diffeomorphisms of Elliptic 3-Manifolds

TL;DR

This paper proves the Smale Conjecture for all elliptic 3-manifolds containing geometrically incompressible Klein bottles and for all lens spaces with m ≥ 3, advancing understanding of their diffeomorphism groups.

Contribution

It confirms the Smale Conjecture for broad classes of elliptic 3-manifolds, including many with Klein bottles and lens spaces, and analyzes the homotopy types of their diffeomorphism and fiber-preserving diffeomorphism groups.

Findings

Smale Conjecture holds for elliptic 3-manifolds with Klein bottles.

Smale Conjecture holds for lens spaces L(m,q) with m ≥ 3.

The space of Seifert fiberings is contractible except for known exceptions.

Abstract

The elliptic 3-manifolds are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, that is, those that have finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to the diffeomorphism group of M 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. Our main results are 1. The Smale Conjecture holds for all elliptic 3-manifolds containing geometrically incompressible Klein bottles. These include all quaternionic and prism manifolds. 2. The Smale Conjecture holds for all lens spaces L(m,q) with m at least 3. These results complete the Smale Conjecture for all cases except the 3-dimensional real projective space and those admitting a Seifert fibering over the 2-sphere with three exceptional fibers of types (2,3,3), (2,3,4), or (2,3,5). The technical work needed for these results includes the result that if V is a Haken Seifert-fibered 3-manifold, then apart from a small list of known exceptions, the inclusion from the space of fiber-preserving diffeomorphisms of V to the full diffeomorphism group is a homotopy equivalence. This has as a consequence: 3. The space of Seifert fiberings of V has contractible components, and apart from a small list of known exceptions, is contractible. Considerable foundational and background material on diffeomorphism groups is included.