Progress 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 -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 -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.

Autorentext

Nikolai Saveliev, University of Miami, Florida, USA.

Inhalt
PrefaceIntroductionGlossary 1 Heegaard splittings1.1 Introduction1.2 Existence of Heegaard splittings1.3 Stable equivalence of Heegaard splittings1.4 The mapping class group1.5 Manifolds of Heegaard genus 11.6 Seifert manifolds1.7 Heegaard diagrams 2 Dehn surgery2.1 Knots and links in 3-manifolds2.2 Surgery on links in S32.3 Surgery description of lens spaces and Seifert manifolds2.4 Surgery and 4-manifolds 3 Kirby calculus3.1 The linking number3.2 Kirby moves3.3 The linking matrix3.4 Reversing orientation 4 Even surgeries 5 Review of 4-manifolds5.1 Definition of the intersection form 5.2 The unimodular integral forms5.3 Four-manifolds and intersection forms 6 Four-manifolds with boundary6.1 The intersection form6.2 Homology spheres via surgery on knots6.3 Seifert homology spheres6.4 The Rohlin invariant 7 Invariants of knots and links7.1 Seifert surfaces7.2 Seifert matrices7.3 The Alexander polynomial7.4 Other invariants from Seifert surfaces7.5 Knots in homology spheres7.6 Boundary links and the Alexander polynomial 8 Fibered knots8.1 The definition of a fibered knot8.2 The monodromy8.3 More about torus knots8.4 Joins8.5 The monodromy of torus knots8.6 Open book decompositions 9 The Arf-invariant9.1 The Arf-invariant of a quadratic form9.2 The Arf-invariant of a knot 10 Rohlin's theorem10.1 Characteristic surfaces10.2 The definition of q10.3 Representing homology classes by surfaces 11 The Rohlin invariant11.1 Definition of the Rohlin invariant11.2 The Rohlin invariant of Seifert spheres11.3 A surgery formula for the Rohlin invariant11.4 The homology cobordism group 12 The Casson invariant 13 The group SU(2) 14 Representation spaces14.1 The topology of representation spaces 14.2 Irreducible representations14.3 Representations of free groups14.4 Representations of surface groups14.5 Representations for Seifert homology spheres 15 The local properties of representation spaces 16 Casson's invariant for Heegaard splittings16.1 The intersection product 16.2 The orientations16.3 Independence of Heegaard splitting 17 Casson's invariant for knots17.1 Preferred Heegaard splittings17.2 The Casson invariant for knots17.3 The difference cycle17.4 The Casson invariant for boundary links17.5 The Casson invariant of a trefoil 18 An application of the Casson invariant18.1 Triangulating 4-manifolds18.2 Higher-dimensional manifolds 19 The Casson invariant of Seifert manifolds19.1 The space R(p; q; r) 19.2 Calculation of the Casson invariant ConclusionBibliographyIndex