  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.
