Lectures on the Topology Of 3-Manifolds : An Introduction to the Casson Invariant.

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.