《三維流形拓撲學講義（第2版）》是2017年世界圖書出版公司出版的著作，作者是薩維利翁 N. 。
基本介紹
- 書名：《三維流形拓撲學講義（第2版）》
- 作者：薩維利翁 N.
- 出版社：世界圖書出版公司
- 出版時間：2017-04-01
內容簡介,目錄,
內容簡介
《三維流形拓撲學講義》主要介紹低維拓撲和Casson理論，當然也不失適時地引入最近研究進展和課題。包括許多經典材料，如Heegaard分裂、Dehn手術、扭結和連線不變數。從Kirby微積分開始，進一步講述Rohlin定理，直到Casson不變數及其套用，並以簡短介紹蕞新進展作為結束。熟悉基礎代數和微分拓撲，包括基礎群、基本同調理論、橫截性和流形上的龐加萊對偶性的數學和理論物理專業的讀者均可閱讀。
目錄
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 surgeries
4.1 Exercises
5 Review of 4—manifolds
5.1 Definition of the incersection 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 Exercises
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
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—invarianc 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 invariant
11.4 The homology coborclism 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 spaccs
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 splitting
16.4 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 Cassoninvariant for boundary links
175 The Casson invariant of a trefoil
18 An application of the Casson invariant
18.1 Triangulating 4—manifolds
18.2 Higher—dimcnsional manifolds
18.3 Exercises
19 The Casson invariant of Seifert manifolds
19.1 The space R（p，q，r）
19.2 Calculation of the Casson invariant
19.3 Exercises
Conclusion
Bibliography
Index
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 surgeries
4.1 Exercises
5 Review of 4—manifolds
5.1 Definition of the incersection 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 Exercises
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
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—invarianc 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 invariant
11.4 The homology coborclism 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 spaccs
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 splitting
16.4 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 Cassoninvariant for boundary links
175 The Casson invariant of a trefoil
18 An application of the Casson invariant
18.1 Triangulating 4—manifolds
18.2 Higher—dimcnsional manifolds
18.3 Exercises
19 The Casson invariant of Seifert manifolds
19.1 The space R（p，q，r）
19.2 Calculation of the Casson invariant
19.3 Exercises
Conclusion
Bibliography
Index
