4-manifolds / Selman Akbulut.

Format:

Book

Author/Creator:

Akbulut, Selman, 1949- author.

Series:

Oxford graduate texts in mathematics ; 25.

Oxford graduate texts in mathematics ; 25

Language:

English

Subjects (All):

Four-manifolds (Topology).

Physical Description:

xii, 262 pages : illustrations (some color) ; 24 cm.

Edition:

First edition.

Other Title:

Four-manifolds

Place of Publication:

Oxford : Oxford University Press, 2016.

Summary:

4-ManifoIds presents the topology of smooth 4-manifolds in an intuitive self-contained way, developed over a number of years by Professor Akbulut. The text is aimed at graduate students and focuses on then teaching and learning of the subject, giving a direct approach to constructions and theorems which are supplemented by exercises to help the reader work through the details not covered in the proofs. Uniquely, this work contains a hundred color illustrations to demonstrate the rather than providing long-winded and potentially unclear explanations. Key results have been selected that relate to the material discussed and the author has provided examples of how to analyst them with the techniques developed in earlier chapters. Book jacket.

Contents:

1 4-manifold handlebodies 1

1.1 Carving 4

1.2 Sliding handles 6

1.3 Canceling handles 8

1.4 Carving ribbons 10

1.5 Non-orientable handles 14

1.6 Algebraic topology 16

2 Building low-dimensional manifolds 19

2.1 Plumbing 21

2.2 Self plumbing 22

2.3 Some useful diffeomorphisms 23

2.4 Examples 24

2.5 Constructing diffeomorphisms by carving 29

2.6 Shake slice knots 32

2.7 Some classical invariants 33

3 Gluing 4-manifolds along their boundaries 37

3.1 Constructing -M _f N by the upside down method 37

3.2 Constructing -M _f N and M(f) by the cylinder method (roping) 39

3.3 Codimension zero surgery M → M' 42

4 Bundles 43

4.1 T₄ = T₂ × T₂ 43

4.2 Cacime surface 45

4.3 General surface bundles over surfaces 52

4.4 Circle bundles over 3-manifolds 52

4.5 3-manifold bundles over the circle 53

5 3-manifolds 55

5.1 Dehn surgery 56

5.2 From framed links to Heegaard diagrams 57

5.3 Gluing knot complements 59

5.4 Carving 3-manifolds 61

5.5 Kohlin invariant 62

6 Operations 64

6.1 Gluck twisting 64

6.2 Blowing down ribbons 67

6.3 Logarithmic transform 68

6.4 Luttinger surgery 69

6.5 Knot surgery 71

6.6 Rational blowdowns 74

7 Lefschetz fibrations 78

7.1 Elliptic surface E(n) 80

7.2 Dolgachev surfaces 81

7.3 PALFs 84

7.4 ALFs 87

7.5 BLFs 92

8 Sympleetic manifolds 95

8.1 Contact manifolds 96

8.2 Stein manifolds 98

8.3 Eliashberg's characterization of Stein 99

8.4 Convex decomposition of 4-manifolds 101

8.5 M⁴ = |BLF| 103

8.6 Stein = |PALF| 106

8.7 Imbedding Stein to symplectic via PALF 107

8.8 Sympleetic fillings 109

9 Exotic 4-manifolds 112

9.1 Constructing small exotic manifolds 112

9.2 Iterated 0-Whitehead doubles are non-slice 116

9.3 A solution of a conjecture of Zeeman 119

9.4 An exotic R⁴ 119

9.5 An exotic non-orient able closed manifold 120

10 Cork decomposition 123

10.1 Corks 125

10.2 Anticorks 130

10.3 Knotting corks 132

10.4 Plugs 133

11 Covering spaces 138

11.1 Handlebody of coverings 138

11.2 Handlebody of branched coverings 141

11.3 Branched covers along ribbon surfaces 147

12 Complex surfaces 150

12.1 Milnor fibers of isolated singularities 150

12.2 Hypersurfaces that are branched covers of CP² 153

12.3 Handlebody descriptions of V_d 155

12.4 σ(a(b,c) 158

13 Seiberg-Witten invariants 163

13.1 Representations 164

13.2 Action of A*(X) on W<sup>±<sup> 166

13.3 Dirac operator 171

13.4 A special calculation 173

13.5 Seiberg-Witten invariants 174

13.6 S-W when b⁺₂(X) = 1 181

13.7 Blowup formula 182

13.8 S-W for torus surgeries 182

13.9 S-W for manifolds with T³ boundary 183

13.10 S-W for logarithmic transforms 185

13.11 S-W for knot surgery X_K 186

13.12 S-W for S¹ × Y³ 189

13.13 Moduli space near the reducible solution 191

13.14 Almost, complex and sympleetic structures 193

13.15 Antiholomorphic quotients 200

13.16 S-W equations on R × Y³ 201

13.17 Adjunction inequality 202

14 Some applications 206

14.1 10/8 theorem 206

14.2 Cappell-Shaneson homotopy spheres 210

14.3 Flexible contraetible 4-manifolds 219

14.4 Some small closed exotic manifolds 223

14.4.1 An exotic CP²#3CP² 223

14.4.2 An exotic CP²#2CP² 234

14.4.3 Fintushel-Stern reverse engineering 243.

Notes:

Includes bibliographical references (pages 245-260) and index.

ISBN:

0198784864

9780198784869

OCLC:

938360673