Nikolai Saveliev, University of Miami, Florida, USA.

This 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

• PrefaceIntroductionGlossary1 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 diagrams2 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-manifolds3 Kirby calculus3.1 The linking number3.2 Kirby moves3.3 The linking matrix3.4 Reversing orientation4 Even surgeries5 Review of 4-manifolds5.1 Definition of the intersection form 5.2 The unimodular integral forms5.3 Four-manifolds and intersection forms6 Four-manifolds with boundary6.1 The intersection form6.2 Homology spheres via surgery on knots6.3 Seifert homology spheres6.4 The Rohlin invariant7 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 polynomial8 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 decompositions9 The Arf-invariant9.1 The Arf-invariant of a quadratic form9.2 The Arf-invariant of a knot10 Rohlin’s theorem10.1 Characteristic surfaces10.2 The definition of ˜q10.3 Representing homology classes by surfaces11 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 group12 The Casson invariant13 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 spheres15 The local properties of representation spaces16 Casson’s invariant for Heegaard splittings16.1 The intersection product 16.2 The orientations16.3 Independence of Heegaard splitting17 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 trefoil18 An application of the Casson invariant18.1 Triangulating 4-manifolds18.2 Higher-dimensional manifolds19 The Casson invariant of Seifert manifolds19.1 The space R(p; q; r) 19.2 Calculation of the Casson invariantConclusionBibliographyIndex

Nikolai Saveliev, University of Miami, Florida, USA.

This 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

• PrefaceIntroductionGlossary1 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 diagrams2 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-manifolds3 Kirby calculus3.1 The linking number3.2 Kirby moves3.3 The linking matrix3.4 Reversing orientation4 Even surgeries5 Review of 4-manifolds5.1 Definition of the intersection form 5.2 The unimodular integral forms5.3 Four-manifolds and intersection forms6 Four-manifolds with boundary6.1 The intersection form6.2 Homology spheres via surgery on knots6.3 Seifert homology spheres6.4 The Rohlin invariant7 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 polynomial8 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 decompositions9 The Arf-invariant9.1 The Arf-invariant of a quadratic form9.2 The Arf-invariant of a knot10 Rohlin’s theorem10.1 Characteristic surfaces10.2 The definition of ˜q10.3 Representing homology classes by surfaces11 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 group12 The Casson invariant13 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 spheres15 The local properties of representation spaces16 Casson’s invariant for Heegaard splittings16.1 The intersection product 16.2 The orientations16.3 Independence of Heegaard splitting17 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 trefoil18 An application of the Casson invariant18.1 Triangulating 4-manifolds18.2 Higher-dimensional manifolds19 The Casson invariant of Seifert manifolds19.1 The space R(p; q; r) 19.2 Calculation of the Casson invariantConclusionBibliographyIndex