We present a new theory which describes the collection of all tunnels of tunnel number 1 knots in S 3 (up to orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex of the genus-2 handlebody and associated structures. It shows that each knot tunnel is obtained from the tunnel of the trivial knot by a uniquely determined sequence of simple cabling constructions. A cabling construction is determined by a single rational parameter, so there is a corresponding numerical parameterization of all tunnels by sequences of such parameters and some additional data. Up to superficial differences in definition, the final parameter of this sequence is the Scharlemann-Thompson invariant of the tunnel, and the other parameters are the Scharlemann-Thompson invariants of the intermediate tunnels produced by the constructions. We calculate the parameter sequences for tunnels of 2-bridge knots. The theory extends easily to links, and to allow equivalence of tunnels by homeomorphisms that may be orientation-reversing. 57M25
If M is a compact orientable irreducible sufficiently large 3-manifold, then the mapping class group %?(M) contains a subgroup of finite index which is the fundamental group of a finite aspherical CW-complex. If in addition the boundary of M is incompressible, then βf(M) contains a subgroup of finite index which is a duality group. For many cases, the virtual cohomological dimension of β?(M) is calculated.
Preface ix Chapter 1. Introduction 1.1. Motivation 1.2. The main theorems for Haken 3-manifolds 1.3. The main theorems for reducible 3-manifolds 1.4. Examples Chapter 2. Johannson's Characteristic Submanifold Theory 2.1. Fibered 3-manifolds 2.2. Boundary patterns 2.3. Admissible maps and mapping class groups 2.4. Essential maps and useful boundary patterns 2.5. The classical theorems 2.6. Exceptional fibered 3-manifolds 2.7. Vertical and horizontal surfaces and maps 2.8. Fiber-preserving maps 2.9. The characteristic submanifold 2.10. Examples of characteristic submanifolds 2.11. The Classification Theorem 2.12. Miscellaneous topological results Chapter 3. Relative Compression Bodies and Cores 3.1. Relative compression bodies 3.2. Minimally imbedded relative compression bodies 3.3. The maximal incompressible core 3.4. Normally imbedded relative compression bodies 3.5. The normal core and the useful core Chapter 4. Homotopy Types 4.1. Homotopy equivalences preserve usefulness 4.2. Finiteness of homotopy types Chapter 5. Pared 3-Manifolds 5.1. Definitions and basic properties 5.2. The topology of pared manifolds 5.3. The characteristic submanifold of a pared manifold Chapter 6. Small 3-Manifolds 6.1. Small manifolds and small pared manifolds 6.2. Small pared homotopy types v vi CONTENTS Chapter 7. Geometrically Finite Hyperbolic 3-Manifolds 7.1. Basic definitions 7.2. Quasiconformal deformation theory: a review 7.3. The Parameterization Theorem Chapter 8. Statements of Main Theorems 8.1. Statements of Main Topological Theorems 8.2. Statements of Main Hyperbolic Theorem and Corollary 8.3. Derivation of hyperbolic results Chapter 9. The Case When There Is a Compressible Free Side 9.1. Algebraic lemmas 9.2. The finite-index cases 9.3. The infinite-index cases Chapter 10. The Case When the Boundary Pattern Is Useful 10.1. The homomorphism Ψ 10.2. Realizing homotopy equivalences of I-bundles 10.3. Realizing homotopy equivalences of Seifert-fibered manifolds 10.4. Proof of Main Topological Theorem 2 Chapter 11. Dehn Flips
Margalit and Schleimer found examples of roots of the Dehn twist t C about a nonseparating curve C in a closed orientable surface, that is, homeomorphisms h such that h n = t C in the mapping class group. Our main theorem gives elementary number-theoretic conditions that describe the n for which an n th root of t C exists, given the genus of the surface. Among its applications, we show that n must be odd, that the Margalit-Schleimer roots achieve the maximum value of n among the roots for a given genus, and that for a given odd n, n th roots exist for all genera greater than (n − 2)(n − 1)/2. We also describe all n th roots having n greater than or equal to the genus.
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