Description
Book SynopsisProgress in low-dimensional topology has been very quick in the last three decades, leading to the solutions of many difficult problems. Among the earlier highlights of this period was Casson's λ-invariant that was instrumental in proving the vanishing of the Rohlin invariant of homotopy 3-spheres. The proof of the three-dimensional Poincaré conjecture has rendered this application moot but hardly made Casson's contribution less relevant: in fact, a lot of modern day topology, including a multitude of Floer homology theories, can be traced back to his λ-invariant. The principal goal of this book, now in its second revised edition, remains providing an introduction to the low-dimensional topology and Casson's theory; it also reaches out, when appropriate, to more recent research topics. The book covers some classical material, such as Heegaard splittings, Dehn surgery, and invariants of knots and links. It then proceeds through the Kirby calculus and Rohlin's theorem to Casson's invariant and its applications, and concludes with a brief overview of recent developments. The book will be accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic and differential topology, including the fundamental group, basic homology theory, transversality, and Poincaré duality on manifolds.
Trade ReviewThis is an excellent introduction to the Rokhlin and Casson invariants for homology 3-spheres [...], and in particular also to the necessary background material from the theory of 3- and 4-manifolds [...], so the book may serve also as a reasonable short and efficient introduction to some important parts of low-dimensional topology. It grew out of a course for second year graduate students and concentrates 19 lectures on less than 200 pages, including also a glossary on back-ground material from algebraic topology, a collection of exercises, open problems and comments on recent developments [...] To conclude, the author has succeeded in presenting a lot of material in a clear and efficient way, and the book is interesting and stimulating to read. Birge Zimmermann-Huisgen, Zentralblatt MATH
Table of ContentsPreface 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 1.7 Heegaard diagrams 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 8.6 Open book decompositions 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 ˜q 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 for 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 boundary links 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 Bibliography Index
