The wild world of 4-manifolds:
"This book offers a panorama of the topology of simply-connected smooth manifolds of dimension four. Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but small enough so that there is no room to undo the wildness. For example, only manifol...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, R.I.
American Mathematical Society
2005
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "This book offers a panorama of the topology of simply-connected smooth manifolds of dimension four. Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but small enough so that there is no room to undo the wildness. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today." "The structure of the book is modular, organized into a main track of about 200 pages, augmented by extensive notes at the end of each chapter, where many extra details, proofs and developments are presented. To help the reader, the text is peppered with over 250 illustrations and has an extensive index."--BOOK JACKET. |
| Beschreibung: | Includes bibliographical references (p. 567-585) and index |
| Beschreibung: | xvii, 609 p. ill. 27 cm |
| ISBN: | 0821837494 9781470468613 |
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| 100 | 1 | |a Scorpan, Alexandru |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The wild world of 4-manifolds |c Alexandru Scorpan |
| 246 | 1 | 3 | |a Wild world of four-manifolds |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 2005 | |
| 300 | |a xvii, 609 p. |b ill. |c 27 cm | ||
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| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references (p. 567-585) and index | ||
| 520 | 1 | |a "This book offers a panorama of the topology of simply-connected smooth manifolds of dimension four. Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but small enough so that there is no room to undo the wildness. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today." "The structure of the book is modular, organized into a main track of about 200 pages, augmented by extensive notes at the end of each chapter, where many extra details, proofs and developments are presented. To help the reader, the text is peppered with over 250 illustrations and has an extensive index."--BOOK JACKET. | |
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| 650 | 7 | |a Dimension 4 |2 swd | |
| 650 | 7 | |a Manifolds |2 gtt | |
| 650 | 7 | |a Mannigfaltigkeit |2 swd | |
| 650 | 4 | |a Variétés topologiques à 4 dimensions | |
| 650 | 4 | |a Four-manifolds (Topology) | |
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Datensatz im Suchindex
| _version_ | 1814781893689737216 |
|---|---|
| adam_text |
Contents
Preview vii
Contents of the Notes xv
Introduction 1
Front matter 5
Part I. Background Scenery
Contents of Part I 25
Chapter 1. Higher Dimensions and the ft Cobordism Theorem 27
1.1. The statement of the theorem. 28
1.2. Handle decompositions. 32
1.3. Handle moves. 40
1.4. Outline of proof. 43
1.5. The Whitney trick. 45
1.6. Low and high handles; handle trading. 47
1.7. Notes. 54
Chapter 2. Topological 4 Manifolds and /i Cobordisms 69
2.1. Casson handles. 70
2.2. The topological fo cobordism theorem. 80
2.3. Homology 3 spheres bound fake 4 balls. 83
2.4. Smooth failure: the twisted cork. 89
2.5. Notes. 91
xi
xii Contents
Part II. Smooth 4 Manifolds and Intersection Forms
Contents of Part II 107
Chapter 3. Getting Acquainted with Intersection Forms 111
3.1. Preparation: representing homology by surfaces. 112
3.2. Intersection forms. 115
3.3. Essential example: the K3 surface. 127
3.4. Notes. 134
Chapter 4. Intersection Forms and Topology 139
4.1. Whitehead's theorem and homotopy type. 140
4.2. Wall's theorems and /z cobordisms. 149
4.3. Intersection forms and characteristic classes. 160
4.4. Rokhlin's theorem and characteristic elements. 168
4.5. Notes. 173
Chapter 5. Classifications and Counterclassifications 237
5.1. Serre's algebraic classification of forms. 238
5.2. Freedman's topological classification. 239
5.3. Donaldson's smooth exclusions. 243
5.4. Byproducts: exotic R4's. 250
5.5. Notes. 260
Part III. A Survey of Complex Surfaces
Contents of Part III 273
Chapter 6. Running through Complex Geometry 275
6.1. Surfaces. 275
6.2. Curves on surfaces. 277
6.3. Line bundles. 278
6.4. Notes. 283
Chapter 7. The Enriques Kodaira Classification 285
7.1. Blow down till nef. 286
7.2. How nef: numerical dimension. 292
7.3. Alternative: Kodaira dimension. 294
7.4. The Kahler case. 295
7.5. Complex versus diffeomorphic. 296
Contents xiii
7.6. Notes. 299
Chapter 8. Elliptic Surfaces 301
8.1. The rational elliptic surface. 302
8.2. Fiber sums. 306
8.3. Logarithmic transformations. 310
8.4. Topological classification. 314
8.5. Notes. 317
Part IV. Gauge Theory on 4 Manifolds
Contents of Part IV 327
Chapter 9. Prelude, and the Donaldson Invariants 331
9.1. Prelude. 332
9.2. Bundles, connections, curvatures. 333
9.3. We are special: self duality. 350
9.4. The Donaldson invariants. 353
9.5. Notes. 357
Chapter 10. The Seiberg Witten Invariants 375
10.1. Almost complex structures. 376
10.2. Spinc structures and spinors. 382
10.3. Definition of the Seiberg Witten invariants. 396
10.4. Main results and properties. 404
10.5. Invariants of symplectic manifolds. 409
10.6. Invariants of complex surfaces. 412
10.7. Notes. 415
Chapter 11. The Minimum Genus of Embedded Surfaces 481
11.1. Before gauge theory: Kervaire Milnor. 482
11.2. Enter the hero: the adjunction inequality. 486
11.3. Digression: the happy case of 3 manifolds. 491
11.4. Notes. 496
Chapter 12. Wildness Unleashed: The Fintushel Stern Surgery 531
12.1. Gluing results in Seiberg Witten theory. 532
12.2. Review: the Alexander polynomial of a knot. 539
12.3. The knot surgery. 541
12.4. Applications. 545
xiv Contents
12.5. Notes. 547
Epilogue 557
List of Figures and Tables 559
Bibliography 567
Index 587 |
| adam_txt |
Contents
Preview vii
Contents of the Notes xv
Introduction 1
Front matter 5
Part I. Background Scenery
Contents of Part I 25
Chapter 1. Higher Dimensions and the ft Cobordism Theorem 27
1.1. The statement of the theorem. 28
1.2. Handle decompositions. 32
1.3. Handle moves. 40
1.4. Outline of proof. 43
1.5. The Whitney trick. 45
1.6. Low and high handles; handle trading. 47
1.7. Notes. 54
Chapter 2. Topological 4 Manifolds and /i Cobordisms 69
2.1. Casson handles. 70
2.2. The topological fo cobordism theorem. 80
2.3. Homology 3 spheres bound fake 4 balls. 83
2.4. Smooth failure: the twisted cork. 89
2.5. Notes. 91
xi
xii Contents
Part II. Smooth 4 Manifolds and Intersection Forms
Contents of Part II 107
Chapter 3. Getting Acquainted with Intersection Forms 111
3.1. Preparation: representing homology by surfaces. 112
3.2. Intersection forms. 115
3.3. Essential example: the K3 surface. 127
3.4. Notes. 134
Chapter 4. Intersection Forms and Topology 139
4.1. Whitehead's theorem and homotopy type. 140
4.2. Wall's theorems and /z cobordisms. 149
4.3. Intersection forms and characteristic classes. 160
4.4. Rokhlin's theorem and characteristic elements. 168
4.5. Notes. 173
Chapter 5. Classifications and Counterclassifications 237
5.1. Serre's algebraic classification of forms. 238
5.2. Freedman's topological classification. 239
5.3. Donaldson's smooth exclusions. 243
5.4. Byproducts: exotic R4's. 250
5.5. Notes. 260
Part III. A Survey of Complex Surfaces
Contents of Part III 273
Chapter 6. Running through Complex Geometry 275
6.1. Surfaces. 275
6.2. Curves on surfaces. 277
6.3. Line bundles. 278
6.4. Notes. 283
Chapter 7. The Enriques Kodaira Classification 285
7.1. Blow down till nef. 286
7.2. How nef: numerical dimension. 292
7.3. Alternative: Kodaira dimension. 294
7.4. The Kahler case. 295
7.5. Complex versus diffeomorphic. 296
Contents xiii
7.6. Notes. 299
Chapter 8. Elliptic Surfaces 301
8.1. The rational elliptic surface. 302
8.2. Fiber sums. 306
8.3. Logarithmic transformations. 310
8.4. Topological classification. 314
8.5. Notes. 317
Part IV. Gauge Theory on 4 Manifolds
Contents of Part IV 327
Chapter 9. Prelude, and the Donaldson Invariants 331
9.1. Prelude. 332
9.2. Bundles, connections, curvatures. 333
9.3. We are special: self duality. 350
9.4. The Donaldson invariants. 353
9.5. Notes. 357
Chapter 10. The Seiberg Witten Invariants 375
10.1. Almost complex structures. 376
10.2. Spinc structures and spinors. 382
10.3. Definition of the Seiberg Witten invariants. 396
10.4. Main results and properties. 404
10.5. Invariants of symplectic manifolds. 409
10.6. Invariants of complex surfaces. 412
10.7. Notes. 415
Chapter 11. The Minimum Genus of Embedded Surfaces 481
11.1. Before gauge theory: Kervaire Milnor. 482
11.2. Enter the hero: the adjunction inequality. 486
11.3. Digression: the happy case of 3 manifolds. 491
11.4. Notes. 496
Chapter 12. Wildness Unleashed: The Fintushel Stern Surgery 531
12.1. Gluing results in Seiberg Witten theory. 532
12.2. Review: the Alexander polynomial of a knot. 539
12.3. The knot surgery. 541
12.4. Applications. 545
xiv Contents
12.5. Notes. 547
Epilogue 557
List of Figures and Tables 559
Bibliography 567
Index 587 |
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| spelling | Scorpan, Alexandru Verfasser aut The wild world of 4-manifolds Alexandru Scorpan Wild world of four-manifolds Providence, R.I. American Mathematical Society 2005 xvii, 609 p. ill. 27 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (p. 567-585) and index "This book offers a panorama of the topology of simply-connected smooth manifolds of dimension four. Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but small enough so that there is no room to undo the wildness. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today." "The structure of the book is modular, organized into a main track of about 200 pages, augmented by extensive notes at the end of each chapter, where many extra details, proofs and developments are presented. To help the reader, the text is peppered with over 250 illustrations and has an extensive index."--BOOK JACKET. Differentialtopologie swd Dimension 4 swd Manifolds gtt Mannigfaltigkeit swd Variétés topologiques à 4 dimensions Four-manifolds (Topology) Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 gnd rswk-swf Dimension 4 (DE-588)4338676-3 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 s Dimension 4 (DE-588)4338676-3 s Differentialgeometrie (DE-588)4012248-7 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014167360&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Scorpan, Alexandru The wild world of 4-manifolds Differentialtopologie swd Dimension 4 swd Manifolds gtt Mannigfaltigkeit swd Variétés topologiques à 4 dimensions Four-manifolds (Topology) Differentialgeometrie (DE-588)4012248-7 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Dimension 4 (DE-588)4338676-3 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4037379-4 (DE-588)4338676-3 |
| title | The wild world of 4-manifolds |
| title_alt | Wild world of four-manifolds |
| title_auth | The wild world of 4-manifolds |
| title_exact_search | The wild world of 4-manifolds |
| title_exact_search_txtP | The wild world of 4-manifolds |
| title_full | The wild world of 4-manifolds Alexandru Scorpan |
| title_fullStr | The wild world of 4-manifolds Alexandru Scorpan |
| title_full_unstemmed | The wild world of 4-manifolds Alexandru Scorpan |
| title_short | The wild world of 4-manifolds |
| title_sort | the wild world of 4 manifolds |
| topic | Differentialtopologie swd Dimension 4 swd Manifolds gtt Mannigfaltigkeit swd Variétés topologiques à 4 dimensions Four-manifolds (Topology) Differentialgeometrie (DE-588)4012248-7 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Dimension 4 (DE-588)4338676-3 gnd |
| topic_facet | Differentialtopologie Dimension 4 Manifolds Mannigfaltigkeit Variétés topologiques à 4 dimensions Four-manifolds (Topology) Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014167360&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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