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| Product Code: | FOURMAN.S |
| List Price: | $95.00 |
| MAA Member Price: | $85.50 |
| AMS Member Price: | $76.00 |
| eBook ISBN: | 978-1-4704-2489-3 |
| Product Code: | FOURMAN.E |
| List Price: | $80.00 |
| MAA Member Price: | $72.00 |
| AMS Member Price: | $64.00 |
| Softcover ISBN: | 978-1-4704-6861-3 |
| eBook ISBN: | 978-1-4704-2489-3 |
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| List Price: | $175.00 $135.00 |
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-
Book Details2005; 614 ppMSC: Primary 57; Secondary 14; 32
This is 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 too small to have room to undo them. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today.
The first part of the book puts things in context with a survey of higher dimensions and of topological 4-manifolds. The second part investigates the main invariant of a 4-manifold—the intersection form—and its interaction with the topology of the manifold. The third part reviews complex surfaces as an important source of examples. The fourth and final part of the book presents gauge theory. This differential-geometric method has brought to light the unwieldy nature of smooth 4-manifolds; and although the method brings new insights, it has raised more questions than answers.
The structure of the book is modular and organized into a main track of approximately 200 pages, which is augmented with copious notes at the end of each chapter, presenting many extra details, proofs, and developments. To help the reader, the text is peppered with over 250 illustrations and has an extensive index.
ReadershipGraduate students and research mathematicians interested in low-dimensional topology.
-
Table of Contents
-
Front Cover
-
Dedication
-
Preview
-
Contents
-
Contents of the Notes
-
Introduction
-
Front matter
-
Part I: Background Scenery
-
Contents of Part I
-
Chapter 1: Higher Dimensions and the h–Cobordism Theorem
-
1.1. The statement of the theorem
-
1.2. Handle decompositions
-
1.3. Handle moves
-
1.4. Outline of proof
-
1.5. The Whitney trick
-
1.6. Low and high handles; handle trading
-
1.7. Notes
-
Chapter 2: Topological 4–Manifoldsand h–Cobordisms
-
2.1. Casson handles
-
2.2. The topological h–cobordism theorem
-
2.3. Homology 3–spheres bound fake 4–balls
-
2.4. Smooth failure: the twisted cork
-
2.5. Notes
-
Part II: Smooth 4–Manifolds and Intersection Forms
-
Contents of Part II
-
Chapter 3: Getting Acquainted with Intersection Forms
-
3.1. Preparation: representing homology by surfaces
-
3.2. Intersection forms
-
3.3. Essential example: the K3 surface
-
3.4. Notes
-
Chapter 4: Intersection Forms and Topology
-
4.1. Whitehead’s theorem and homotopy type
-
4.2. Wall’s theorems and h–cobordisms
-
4.3. Intersection forms and characteristic classes
-
4.4. Rokhlin’s theorem and characteristic elements
-
4.5. Notes
-
Chapter 5: Classifications and Counterclassifications
-
5.1. Serre’s algebraic classification of forms
-
5.2. Freedman’s topological classification
-
5.3. Donaldson’s smooth exclusions
-
5.4. Byproducts: exotic R4 ’s
-
5.5. Notes
-
Part III: A Survey of Complex Surfaces
-
Contents of Part III
-
Chapter 6: Running through Complex Geometry
-
6.1. Surfaces
-
6.2. Curves on surfaces
-
6.3. Line bundles
-
6.4. Notes
-
Chapter 7: The Enriques–Kodaira Classification
-
7.1. Blow-down till nef
-
7.2. How nef: numerical dimension
-
7.3. Alternative: Kodaira dimension
-
7.4. The Kähler case
-
7.5. Complex versus diffeomorphic
-
7.6. Notes
-
Chapter 8: Elliptic Surfaces
-
8.1. The rational elliptic surface
-
8.2. Fiber-sums
-
8.3. Logarithmic transformations
-
8.4. Topological classification
-
8.5. Notes
-
Part IV: Gauge Theory on 4–Manifolds
-
Contents of Part IV
-
Chapter 9: Prelude, and the Donaldson Invariants
-
9.1. Prelude
-
9.2. Bundles, connections, curvatures
-
9.3. We are special: self-duality
-
9.4. The Donaldson invariants
-
9.5. Notes
-
Chapter 10: The Seiberg–Witten Invariants
-
10.1. Almost-complex structures
-
10.2. SpinC structures and spinors
-
10.3. Definition of the Seiberg–Witten invariants
-
10.4. Main results and properties
-
10.5. Invariants of symplectic manifolds
-
10.6. Invariants of complex surfaces
-
10.7. Notes
-
Chapter 11: The Minimum Genus of Embedded Surfaces
-
11.1. Before gauge theory: Kervaire–Milnor
-
11.2. Enter the hero: the adjunction inequality
-
11.3. Digression: the happy case of 3–manifolds
-
11.4. Notes
-
Chapter 12: Wildness Unleashed: The Fintushel–Stern Surgery
-
12.1. Gluing results in Seiberg–Witten theory
-
12.2. Review: the Alexander polynomial of a knot
-
12.3. The knot surgery
-
12.4. Applications
-
12.5. Notes
-
Epilogue
-
List of Figures and Tables
-
Bibliography
-
Index
-
Errata
-
Back Cover
-
-
Additional Material
-
Reviews
-
The book gives an excellent overview of 4-manifolds, with many figures and historical notes. Graduate students, nonexperts, and experts alike will enjoy browsing through it.
Robion C. Kirby, University of California, Berkeley -
What a wonderful book! I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds.
MAA Reviews -
The author records many spectacular results in the subject ... (the author) gives the reader a taste of the techniques involved in the proofs, geometric topology, gauge theory and complex and symplectic structures.
The book has a large and up-to-date collection of references for the reader wishing to get a more detailed or rigorous knowledge of a specific topic. The exposition is user-friendly, with a large number of illustrations and examples.
Mathematical Reviews
-
-
RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
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This is 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 too small to have room to undo them. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today.
The first part of the book puts things in context with a survey of higher dimensions and of topological 4-manifolds. The second part investigates the main invariant of a 4-manifold—the intersection form—and its interaction with the topology of the manifold. The third part reviews complex surfaces as an important source of examples. The fourth and final part of the book presents gauge theory. This differential-geometric method has brought to light the unwieldy nature of smooth 4-manifolds; and although the method brings new insights, it has raised more questions than answers.
The structure of the book is modular and organized into a main track of approximately 200 pages, which is augmented with copious notes at the end of each chapter, presenting many extra details, proofs, and developments. To help the reader, the text is peppered with over 250 illustrations and has an extensive index.
Graduate students and research mathematicians interested in low-dimensional topology.
-
Front Cover
-
Dedication
-
Preview
-
Contents
-
Contents of the Notes
-
Introduction
-
Front matter
-
Part I: Background Scenery
-
Contents of Part I
-
Chapter 1: Higher Dimensions and the h–Cobordism Theorem
-
1.1. The statement of the theorem
-
1.2. Handle decompositions
-
1.3. Handle moves
-
1.4. Outline of proof
-
1.5. The Whitney trick
-
1.6. Low and high handles; handle trading
-
1.7. Notes
-
Chapter 2: Topological 4–Manifoldsand h–Cobordisms
-
2.1. Casson handles
-
2.2. The topological h–cobordism theorem
-
2.3. Homology 3–spheres bound fake 4–balls
-
2.4. Smooth failure: the twisted cork
-
2.5. Notes
-
Part II: Smooth 4–Manifolds and Intersection Forms
-
Contents of Part II
-
Chapter 3: Getting Acquainted with Intersection Forms
-
3.1. Preparation: representing homology by surfaces
-
3.2. Intersection forms
-
3.3. Essential example: the K3 surface
-
3.4. Notes
-
Chapter 4: Intersection Forms and Topology
-
4.1. Whitehead’s theorem and homotopy type
-
4.2. Wall’s theorems and h–cobordisms
-
4.3. Intersection forms and characteristic classes
-
4.4. Rokhlin’s theorem and characteristic elements
-
4.5. Notes
-
Chapter 5: Classifications and Counterclassifications
-
5.1. Serre’s algebraic classification of forms
-
5.2. Freedman’s topological classification
-
5.3. Donaldson’s smooth exclusions
-
5.4. Byproducts: exotic R4 ’s
-
5.5. Notes
-
Part III: A Survey of Complex Surfaces
-
Contents of Part III
-
Chapter 6: Running through Complex Geometry
-
6.1. Surfaces
-
6.2. Curves on surfaces
-
6.3. Line bundles
-
6.4. Notes
-
Chapter 7: The Enriques–Kodaira Classification
-
7.1. Blow-down till nef
-
7.2. How nef: numerical dimension
-
7.3. Alternative: Kodaira dimension
-
7.4. The Kähler case
-
7.5. Complex versus diffeomorphic
-
7.6. Notes
-
Chapter 8: Elliptic Surfaces
-
8.1. The rational elliptic surface
-
8.2. Fiber-sums
-
8.3. Logarithmic transformations
-
8.4. Topological classification
-
8.5. Notes
-
Part IV: Gauge Theory on 4–Manifolds
-
Contents of Part IV
-
Chapter 9: Prelude, and the Donaldson Invariants
-
9.1. Prelude
-
9.2. Bundles, connections, curvatures
-
9.3. We are special: self-duality
-
9.4. The Donaldson invariants
-
9.5. Notes
-
Chapter 10: The Seiberg–Witten Invariants
-
10.1. Almost-complex structures
-
10.2. SpinC structures and spinors
-
10.3. Definition of the Seiberg–Witten invariants
-
10.4. Main results and properties
-
10.5. Invariants of symplectic manifolds
-
10.6. Invariants of complex surfaces
-
10.7. Notes
-
Chapter 11: The Minimum Genus of Embedded Surfaces
-
11.1. Before gauge theory: Kervaire–Milnor
-
11.2. Enter the hero: the adjunction inequality
-
11.3. Digression: the happy case of 3–manifolds
-
11.4. Notes
-
Chapter 12: Wildness Unleashed: The Fintushel–Stern Surgery
-
12.1. Gluing results in Seiberg–Witten theory
-
12.2. Review: the Alexander polynomial of a knot
-
12.3. The knot surgery
-
12.4. Applications
-
12.5. Notes
-
Epilogue
-
List of Figures and Tables
-
Bibliography
-
Index
-
Errata
-
Back Cover
-
The book gives an excellent overview of 4-manifolds, with many figures and historical notes. Graduate students, nonexperts, and experts alike will enjoy browsing through it.
Robion C. Kirby, University of California, Berkeley -
What a wonderful book! I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds.
MAA Reviews -
The author records many spectacular results in the subject ... (the author) gives the reader a taste of the techniques involved in the proofs, geometric topology, gauge theory and complex and symplectic structures.
The book has a large and up-to-date collection of references for the reader wishing to get a more detailed or rigorous knowledge of a specific topic. The exposition is user-friendly, with a large number of illustrations and examples.
Mathematical Reviews
