Lectures on the topology of 3-manifolds : an introduction to the Casson invariant / Nikolai Saveliev.

Format:

Book

Author/Creator:

Saveliev, Nikolai, 1966-

Series:

De Gruyter textbook

Language:

English

Subjects (All):

Manifolds (Mathematics).

Three-manifolds (Topology).

Physical Description:

1 online resource (212 p.)

Edition:

1st ed.

Place of Publication:

Berlin ; New York : Walter de Gruyter, 1999.

Language Note:

English

Summary:

No detailed description available for "Lectures on the Topology of 3-Manifolds".

Contents:

Intro

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.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references (pages [186]-195) and index.

Description based on print version record.

ISBN:

9783110806359

3110806355

OCLC:

868974008