Lectures on the Topology of 3-Manifolds
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Lectures on the Topology of 3-Manifolds

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eBook - PDF

Lectures on the Topology of 3-Manifolds

Book details
Table of contents
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Information

Publisher
De Gruyter
Year
2012
ISBN
9783110806359
Edition
1
Topic
Mathematics
Subtopic
Mathematics General
Index
Mathematics

Table of contents

  1. Preface
  2. Introduction
  3. Glossary
  4. 1 Heegaard splittings
  5. 1.1 Introduction
  6. 1.2 Existence of Heegaard splittings
  7. 1.3 Stable equivalence of Heegaard splittings
  8. 1.4 The mapping class group
  9. 1.5 Manifolds of Heegaard genus ≤ 1
  10. 1.6 Seifert manifolds
  11. 2 Dehn surgery
  12. 2.1 Knots and links in 3-manifolds
  13. 2.2 Surgery on links in S3
  14. 2.3 Surgery description of lens spaces and Seifert manifolds
  15. 2.4 Surgery and 4-manifolds
  16. 3 Kirby calculus
  17. 3.1 The linking number
  18. 3.2 Kirby moves
  19. 3.3 The linking matrix
  20. 3.4 Reversing orientation
  21. 4 Even surgeries
  22. 5 Review of 4-manifolds
  23. 5.1 Definition of the intersection form
  24. 5.2 The unimodular integral forms
  25. 5.3 Four-manifolds and intersection forms
  26. 6 Four-manifolds with boundary
  27. 6.1 The intersection form
  28. 6.2 Homology spheres via surgery on knots
  29. 6.3 Seifert homology spheres
  30. 6.4 The Rohlin invariant
  31. 7 Invariants of knots and links
  32. 7.1 Seifert surfaces
  33. 7.2 Seifert matrices
  34. 7.3 The Alexander polynomial
  35. 7.4 Other invariants from Seifert surfaces
  36. 7.5 Knots in homology spheres
  37. 7.6 Boundary links and the Alexander polynomial
  38. 8 Fibered knots
  39. 8.1 The definition of a fibered knot
  40. 8.2 The monodromy
  41. 8.3 More about torus knots
  42. 8.4 Joins
  43. 8.5 The monodromy of torus knots
  44. 9 The Arf-invariant
  45. 9.1 The Arf-invariant of a quadratic form
  46. 9.2 The Arf-invariant of a knot
  47. 10 Rohlin’s theorem
  48. 10.1 Characteristic surfaces
  49. 10.2 The definition of q̃
  50. 10.3 Representing homology classes by surfaces
  51. 11 The Rohlin invariant
  52. 11.1 Definition of the Rohlin invariant
  53. 11.2 The Rohlin invariant of Seifert spheres
  54. 11.3 A surgery formula for the Rohlin invariant
  55. 11.4 The homology cobordism group
  56. 12 The Casson invariant
  57. 13 The group SU(2)
  58. 14 Representation spaces
  59. 14.1 The topology of representation spaces
  60. 14.2 Irreducible representations
  61. 14.3 Representations of free groups
  62. 14.4 Representations of surface groups
  63. 14.5 Representations of Seifert homology spheres
  64. 15 The local properties of representation spaces
  65. 16 Casson’s invariant for Heegaard splittings
  66. 16.1 The intersection product
  67. 16.2 The orientations
  68. 16.3 Independence of Heegaard splitting
  69. 17 Casson’s invariant for knots
  70. 17.1 Preferred Heegaard splittings
  71. 17.2 The Casson invariant for knots
  72. 17.3 The difference cycle
  73. 17.4 The Casson invariant for unlinks
  74. 17.5 The Casson invariant of a trefoil
  75. 18 An application of the Casson invariant
  76. 18.1 Triangulating 4-manifolds
  77. 18.2 Higher-dimensional manifolds
  78. 19 The Casson invariant of Seifert manifolds
  79. 19.1 The space R(p, q, r)
  80. 19.2 Calculation of the Casson invariant
  81. Conclusion
  82. Exercises
  83. Bibliography
  84. Index