tailieunhanh - Đề tài " Homotopy hyperbolic 3manifolds are hyperbolic "

This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This procedure is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-manifold. Theorem . Let N be a closed hyperbolic 3-manifold. | Annals of Mathematics Homotopy hyperbolic 3-manifolds are hyperbolic By David Gabai G. Robert Meyerhoff and Nathaniel Thurston Annals of Mathematics 157 2003 335 431 Homotopy hyperbolic 3-manifolds are hyperbolic By David Gabai G. Robert MEyERHOFF and Nathaniel Thurston 0. Introduction This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This procedure is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-manifold. Theorem . Let N be a closed hyperbolic 3-manifold. Then i If f M N is a homotopy equivalence where M is a closed irreducible 3-manifold then f is homotopic to a homeomorphism. ii If f g M N are homotopic homeomorphisms then f is isotopic to g. iii The space of hyperbolic metrics on N is path connected. Remarks. Under the additional hypothesis that M is hyperbolic conclusion i follows from Mostow s rigidity theorem Mo . Under the hypothesis that N is Haken and not necessarily hyperbolic conclusions i ii follow from Waldhausen Wa . Under the hypothesis that N is both Haken and hyperbolic conclusion iii follows by combination of Mo and Wa . Because non-Haken manifolds are necessarily orientable we will from now on assume that all manifolds under discussion are orientable. Theorem with the added hypothesis that some closed geodesic ỗ c N has a noncoalescable insulator family was proven by Gabai see G . Thus Theorem follows from G and the main technical result of this paper which is Theorem . If ỗ is a shortest geodesic in a closed orientable hyperbolic 3-manifold then ỗ has a non-coalescable insulator family. Remarks. If ỗ is the core of an embedded hyperbolic tube of radius ln 3 2 . then ỗ has a noncoalescable insulator family by Lemma of G . See the Appendix to this paper for a review of insulator theory. 336 DAVID GABAI G. ROBERT MEYERHOFF AND .