Ebook: Lectures on the Topology of 3-Manifolds : an Introduction to the Casson Invariant

This textbook, now in its second revised and extended edition, introduces the topology of 3- and 4-dimensional manifolds. It also considers new developments especially related to the Heegaard Floer and contact homology. The book is accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic topology, including the fundamental group, basic homology theory, and Poincaré duality on manifolds. Read more... 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; 1.7 Heegaard diagrams; 1.8 Exercises; 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; 2.5 Exercises; 3 Kirby calculus; 3.1 The linking number; 3.2 Kirby moves; 3.3 The linking matrix; 3.4 Reversing orientation; 3.5 Exercises. 4 Even surgeries4.1 Exercises; 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; 5.4 Exercises; 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; 6.5 Exercises; 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. 7.7 Exercises8 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; 8.7 Exercises; 9 The Arf-invariant; 9.1 The Arf-invariant of a quadratic form; 9.2 The Arf-invariant of a knot; 9.3 Exercises; 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 invariant11.4 The homology cobordism group; 11.5 Exercises; 12 The Casson invariant; 12.1 Exercises; 13 The group SU (2); 13.1 Exercises; 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; 14.6 Exercises; 15 The local properties of representation spaces; 15.1 Exercises; 16 Casson's invariant for Heegaard splittings; 16.1 The intersection product; 16.2 The orientations. 16.3 Independence of Heegaard splitting16.4 Exercises; 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; 18.3 Exercises; 19 The Casson invariant of Seifert manifolds; 19.1 The space R(S (p, q, r)); 19.2 Calculation of the Casson invariant; 19.3 Exercises; Conclusion; Bibliography; Index