Lecture Notes On Generalized Heegaard Splittings

by Martin Scharlemann, Jennifer Schultens, Toshio Saito

Overview

This book is part of the series of three books arise from lectures organized by Hitoshi Murakami at RIMS, Kyoto University in the summer of 2001. The lecture series was aimed at a broad audience that included many graduate students. Its purpose lay in familiarizing the audience with the basics of 3-manifold theory and introducing some topics of current research. The first portion of the lecture series was devoted to standard topics in the theory of 3-manifolds. The middle portion was devoted to a brief study of Heegaard splittings and generalized Heegaard splittings.In the standard schematic diagram for generalized Heegaard splittings, Heegaard splittings are stacked on top of each other in a linear fashion. This can cause confusion in those cases in which generalized Heegaard splittings possess interesting connectivity properties. Fork complexes were invented in an effort to illuminate some of the more subtle issues arising in the study of generalized Heegaard splittings.

Product Details

ISBN-13:	9789813109117
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/14/2016
Pages:	140
Product dimensions:	6.10(w) x 9.10(h) x 0.60(d)

Table of Contents

Preface v

1 Preliminaries 1

1.1 Manifolds 1

1.2 Simplicial complexes 2

1.3 PL manifolds 5

1.4 Fundamental definitions 8

2 Definition and examples of Heegaard splittings 13

2.1 Definitions 13

2.2 Examples 22

2.2.1 M = S³ 22

2.2.2 M = S² x S¹ 24

2.2.3 Lens spaces 25

2.3 Bridge position and tunnel number 28

3 Properties of Heegaard splittings 31

3.1 Reducibility, weak reducibility and stabilization 31

3.2 Haken's Lemma 34

4 Two theorems on Heegaard splittings 49

4.1 Waldhausen's Theorem 49

4.2 Reidemeister-Singer Theorem 88

5 Generalized Heegaard splittings 91

5.1 Definitions 91

5.2 Thin generalized Heegaard splittings 95

5.3 Examples 107

5.3.1 M = F₉ x [0,1] 107

5.3.2 M = F₉ x S¹ 108

5.4 Amalgamation 114

5.5 Boundary stabilization and the stabilization problem 118

Bibliography 123

Index 129