Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant (De Gruyter Textbook) (De Gruyter Textbook)

All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typesetting using the author's TgX files: I. Zimmermann, Freiburg. Printing and Binding: WB-Druck GmbH & Co., Rieden/Allgäu. sification, Rohlin invariant, SU(2)-representation spaces, twisted cohomology, Casson invariant, etc. Throughout the book, we mention the latest developments whenever it seems appropriate. For example, in the section on 4-manifold topology, we give a review of recent results relating 4-manifolds and unimodular forms, including the "10/8-conjecture" and Donaldson polynomials. The Rohlin invariant gives restrictions on the genus of surfaces embedded in a smooth 4-manifold. When describing this old result, we also survey the results that follow from the Thom conjecture, proved a few years ago by Kronheimer and Mrowka with the help of Seiberg-Witten theory. The topology of 3-manifolds includes a variety of topics not discussed in this book, among which are hyperbolic manifolds, Thurston's geometrization conjecture, incompressible surfaces, prime decompositions of 3-manifolds, and many others. The book has a list of exercises, brief notes on further developments, and a short list of open problems. The book is closely related, in several instances, both in content and method, to the books Akbulut-McCarthy [2] and Fomenko-Matveev [47], from which I have borrowed quite shamelessly. However, it is hoped that the present treatment will serve its purpose of providing an accessible introduction to certain topics in the topology of 3manifolds. Preface Introduction Glossary 1 Heegaard splittings 1.1 Introduction 1.2 Existence of Heegaard splittings 1.3 Stable equivalence of Heegaard splittings 1.4 The mapping class group 1.5 Manifolds of Heegaard genus ≤ 1 1.6 Seifert manifolds 2 Dehn surgery 2.1 Knots and links in 3-manifolds 2.2 Surgery on links in S3 2.3 Surgery description of lens spaces and Seifert manifolds 2.4 Surgery and 4-manifolds 3 Kirby calculus 3.1 The linking number 3.2 Kirby moves 3.3 The linking matrix 3.4 Reversing orientation 4 Even surgeries 5 Review of 4-manifolds 5.1 Definition of the intersection form 5.2 The unimodular integral forms 5.3 Four-manifolds and intersection forms 6 Four-manifolds with boundary 6.1 The intersection form 6.2 Homology spheres via surgery on knots 6.3 Seifert homology spheres 6.4 The Rohlin invariant 7 Invariants of knots and links 7.1 Seifert surfaces 7.2 Seifert matrices 7.3 The Alexander polynomial 7.4 Other invariants from Seifert surfaces 7.5 Knots in homology spheres 7.6 Boundary links and the Alexander polynomial 8 Fibered knots 8.1 The definition of a fibered knot 8.2 The monodromy 8.3 More about torus knots 8.4 Joins 8.5 The monodromy of torus knots 9 The Arf-invariant 9.1 The Arf-invariant of a quadratic form 9.2 The Arf-invariant of a knot 10 Rohlin’s theorem 10.1 Characteristic surfaces 10.2 The definition of q̃ 10.3 Representing homology classes by surfaces 11 The Rohlin invariant 11.1 Definition of the Rohlin invariant 11.2 The Rohlin invariant of Seifert spheres 11.3 A surgery formula for the Rohlin invariant 11.4 The homology cobordism group 12 The Casson invariant 13 The group SU(2) 14 Representation spaces 14.1 The topology of representation spaces 14.2 Irreducible representations 14.3 Representations of free groups 14.4 Representations of surface groups 14.5 Representations of Seifert homology spheres 15 The local properties of representation spaces 16 Casson’s invariant for Heegaard splittings 16.1 The intersection product 16.2 The orientations 16.3 Independence of Heegaard splitting 17 Casson’s invariant for knots 17.1 Preferred Heegaard splittings 17.2 The Casson invariant for knots 17.3 The difference cycle 17.4 The Casson invariant for unlinks 17.5 The Casson invariant of a trefoil 18 An application of the Casson invariant 18.1 Triangulating 4-manifolds 18.2 Higher-dimensional manifolds 19 The Casson invariant of Seifert manifolds 19.1 The space R(p, q, r) 19.2 Calculation of the Casson invariant Conclusion Exercises Bibliography Index

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Progress In Low-dimensional Topology Has Been Very Fast In The Last Two Decades, Leading To The Solutions Of Many Difficult Problems. Among The Highlights Of This Period Are Casson's Results On The Rohlin Invariant Of Homotopy 3-spheres, As Well As His [lambda]-invariant. The Purpose Of This Book Is To Provide A Much-needed Bridge To These Modern Topics. The Book Covers Some Classical Topics, Such As Heegaard Splittings, Dehn Surgery, And Invariants Of Knots And Links. It Proceeds Through The Kirby Calculus And Rohlin's Theorem To Casson's Invariant And Its Applications, And Gives A Brief Sketch Of Links With The Latest Developments In Low-dimensional Topology And Gauge Theory. The Text Will Be Accessible To Graduate Students In Mathematics And Theoretical Physics Familiar With Some Elementary Algebraic Topology, Including The Fundamental Group, Basic Homology Theory, And Poincare Duality On Manifolds.--jacket. 1. Heegaard Splittings -- 2. Dehn Surgery -- 3. Kirby Calculus -- 4. Even Surgeries -- 5. Review Of 4-manifolds -- 6. Four-manifolds With Boundary -- 7. Invariants Of Knots And Links -- 8. Fibered Knots -- 9. The Arf-invariant -- 10. Rohlin's Theorem -- 11. The Rohlin Invariant -- 12. The Casson Invariant -- 13. The Group Su(2) -- 14. Representation Spaces -- 15. The Local Properties Of Representation Spaces -- 16. Casson's Invariant For Heegaard Splittings -- 17. Casson's Invariant For Knots -- 18. An Application Of The Casson Invariant -- 19. The Casson Invariant Of Seifert Manifolds. Nikolaĭ Saveliev. Includes Bibliographical References (p. [186]-195) And Index. 'Progress in low-dimensional topology has been very fast in the last two decades, leading to the solutions of many difficult problems.''Among the highlights of this period are Casson's results on the Rohlin invariant of homotopy 3-spheres, as well as his [lambda]-invariant. The purpose of this book is to provide a much-needed bridge to these modern topics. The book covers some classical topics, such as Heegaard splittings, Dehn surgery, and invariants of knots and links. It proceeds through the Kirby calculus and Rohlin's theorem to Casson's invariant and its applications, and gives a brief sketch of links with the latest developments in low-dimensional topology and gauge theory.''The text will be accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic topology, including the fundamental group, basic homology theory, and Poincare duality on manifolds.'--BOOK JACKET.