By Sungbok Hong, John Kalliongis, Darryl McCullough, J. Hyam Rubinstein
This paintings matters the diffeomorphism teams of 3-manifolds, specifically of elliptic 3-manifolds. those are the closed 3-manifolds that admit a Riemannian metric of continuous optimistic curvature, referred to now to be precisely the closed 3-manifolds that experience a finite primary team. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry workforce of M to its diffeomorphism staff is a homotopy equivalence. the unique Smale Conjecture, for the 3-sphere, used to be confirmed through J. Cerf and A. Hatcher, and N. Ivanov proved the generalized conjecture for plenty of of the elliptic 3-manifolds that include a geometrically incompressible Klein bottle.
The major effects identify the Smale Conjecture for all elliptic 3-manifolds containing geometrically incompressible Klein bottles, and for all lens areas L(m,q) with m at the very least three. extra effects indicate that for a Haken Seifert-fibered three manifold V, the distance of Seifert fiberings has contractible parts, and except a small record of identified exceptions, is contractible. substantial foundational and historical past
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Note that since M may have boundary, it is necessary to use the half-space version of reference  at points of V \@M . W . 1; 1///. At points of @M , the component of each vector in the direction perpendicular to @M is 0, so since E is linear, the extended component is also 0 and therefore the extended vector field is also tangent to the boundary. V; TM/ ! M; TM/. @M S /. 3 (Exponentiation Lemma). Assume that the metric on M is a product near the boundary, and let K be a compact subset of M .
O/. 8. An orbifold homeomorphism of O is a homeomorphism of the underlying topological space of O that is induced by an H -equivariant homee called a lift of the orbifold homeomorphism. An orbifold omorphism of O, diffeomorphism of O is an orbifold homeomorphism for which some and hence e are diffeomorphisms. O/ to be the group of orbifold all lifts to O e by the normal diffeomorphisms. O/ subgroup H . O/. 9. An orbifold W contained in O is called a suborbifold of O if its e is a submanifold. f W ; O/.
Y/; t/. Proof. We first prove that equivariant Riemannian metrics exist. Choose a compact subset C of M that maps surjectively onto M=H under the quotient map. Let W M ! Œ0; 1/ be a compactly supported smooth function which is positive on C . M /. w// : h2H Since is compactly supported, the sum is finite, and since every orbit meets the support of , R0 is positive definite. To check equivariance, let g 2 H . 5 Restriction Theorems for Orbifolds 33 We need to improve the metric near the boundary.
Diffeomorphisms of Elliptic 3-Manifolds by Sungbok Hong, John Kalliongis, Darryl McCullough, J. Hyam Rubinstein