By Sungbok Hong, John Kalliongis, Darryl McCullough, J. Hyam Rubinstein

This paintings issues the diffeomorphism teams of 3-manifolds, specifically of elliptic 3-manifolds. those are the closed 3-manifolds that admit a Riemannian metric of continuing optimistic curvature, referred to now to be precisely the closed 3-manifolds that experience a finite basic workforce. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry staff of M to its diffeomorphism team is a homotopy equivalence. the unique Smale Conjecture, for the 3-sphere, used to be confirmed via J. Cerf and A. Hatcher, and N. Ivanov proved the generalized conjecture for lots of of the elliptic 3-manifolds that comprise a geometrically incompressible Klein bottle.

The major effects determine the Smale Conjecture for all elliptic 3-manifolds containing geometrically incompressible Klein bottles, and for all lens areas L(m,q) with m not less than three. extra effects suggest that for a Haken Seifert-fibered three manifold V, the gap of Seifert fiberings has contractible elements, and except a small record of identified exceptions, is contractible. significant foundational and historical past

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O rel S / ! ˙ rel T / y ! W /; O rel S / . 7 Spaces of Fibered Structures In this section, we examine spaces of fibered structures. 13. Let pW ˙ ! O be a singular fibering. ˙/. 14. A singular fibering pW ˙ ! O is called very good if ˙ chosen to be compact. The main result of this section is the following fibration theorem. 12. Let pW ˙ ! O be a very good singular fibering. ˙ ; T ˙/. ˙/ ! ˙/ is a locally trivial fibering. Here is the basic idea of the proof. F /. It is not obvious that such a choice is uniquely determined, but there is a way to make one when h is sufficiently close to a fiber-preserving diffeomorphism.

Rel T / cross-sections. Proof. 2, it suffices to find local cross-sections at the inclusion iW . 13, there are an open neighborhood U e W e;˙ e / and a continuous map X W U e ! x/ for all j 2U e . W e;T˙ e/ ! W e ; T ˙/ e and x 2 W e. ˙; e T ˙/. E/. U 1 /, the composition TExpa ık ı X W U ! E/ is the desired e;T˙ e /, k carries cross-section for (i). ˙ rel T / cross-section, v giving (ii). t u As in Sect. 4, we have the following immediate corollaries. 8. Let W be a vertical suborbifold of ˙.

M; TM/ for which the map F W U ! Y / is defined and continuous. U /, the function F ı k ı X W U1 ! M rel S / will be the desired crosssection. x/ D x for all x 2 V \ S . x/ for x 2 V \ S . x/ D x for all x 2 S. 1. Let V be a compact submanifold of M . Let S Â @M be a closed neighborhood in @M of S \ @V , and L a neighborhood of V in M . M rel S / ! V; M rel S / is locally trivial. 2. Let V and W be compact submanifolds of M , with W Â V . Let S Â @M a closed neighborhood in @M of S \ @V , and L a neighborhood of V in M .