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Homotopy Equivalences of 3-Manifolds and Deformation Theory of Kleinian Groups
Richard D. Canary, University of Michigan, Ann Arbor, MI, and Darryl McCullough, University of Oklahoma, Norman, OK
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Memoirs of the American Mathematical Society
2004; 218 pp; softcover
Volume: 172
ISBN-10: 0-8218-3549-1
ISBN-13: 978-0-8218-3549-4
List Price: US$75
Individual Members: US$45
Institutional Members: US$60
Order Code: MEMO/172/812
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This text investigates a natural question arising in the topological theory of \(3\)-manifolds, and applies the results to give new information about the deformation theory of hyperbolic \(3\)-manifolds. It is well known that some compact \(3\)-manifolds with boundary admit homotopy equivalences that are not homotopic to homeomorphisms. We investigate when the subgroup \(\mathcal{R}(M)\) of outer automorphisms of \(\pi_1(M)\) which are induced by homeomorphisms of a compact \(3\)-manifold \(M\) has finite index in the group \(\operatorname{Out}(\pi_1(M))\) of all outer automorphisms. This question is completely resolved for Haken \(3\)-manifolds. It is also resolved for many classes of reducible \(3\)-manifolds and \(3\)-manifolds with boundary patterns, including all pared \(3\)-manifolds.

The components of the interior \(\operatorname{GF}(\pi_1(M))\) of the space \(\operatorname{AH}(\pi_1(M))\) of all (marked) hyperbolic \(3\)-manifolds homotopy equivalent to \(M\) are enumerated by the marked homeomorphism types of manifolds homotopy equivalent to \(M\), so one may apply the topological results above to study the topology of this deformation space. We show that \(\operatorname{GF}(\pi_1(M))\) has finitely many components if and only if either \(M\) has incompressible boundary, but no "double trouble," or \(M\) has compressible boundary and is "small." (A hyperbolizable \(3\)-manifold with incompressible boundary has double trouble if and only if there is a thickened torus component of its characteristic submanifold which intersects the boundary in at least two annuli.) More generally, the deformation theory of hyperbolic structures on pared manifolds is analyzed.

Some expository sections detail Johannson's formulation of the Jaco-Shalen-Johannson characteristic submanifold theory, the topology of pared \(3\)-manifolds, and the deformation theory of hyperbolic \(3\)-manifolds. An epilogue discusses related open problems and recent progress in the deformation theory of hyperbolic \(3\)-manifolds.

Readership

Graduate students and research mathematicians interested in geometry and topology.

Table of Contents

  • Introduction
  • Johannson's characteristic submanifold theory
  • Relative compression bodies and cores
  • Homotopy types
  • Pared 3-manifolds
  • Small 3-manifolds
  • Geometrically finite hyperbolic 3-manifolds
  • Statements of main theorems
  • The case when there is a compressible free side
  • The case when the boundary pattern is useful
  • Dehn flips
  • Finite index realization for reducible 3-manifolds
  • Epilogue
  • Bibliography
  • Index
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