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Softcover ISBN:  9781470468613 
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Product Code:  FOURMAN.S.B 
List Price:  $175.00 $135.00 
MAA Member Price:  $157.50 $121.50 
AMS Member Price:  $140.00 $108.00 
Softcover ISBN:  9781470468613 
Product Code:  FOURMAN.S 
List Price:  $95.00 
MAA Member Price:  $85.50 
AMS Member Price:  $76.00 
eBook ISBN:  9781470424893 
Product Code:  FOURMAN.E 
List Price:  $80.00 
MAA Member Price:  $72.00 
AMS Member Price:  $64.00 
Softcover ISBN:  9781470468613 
eBook ISBN:  9781470424893 
Product Code:  FOURMAN.S.B 
List Price:  $175.00 $135.00 
MAA Member Price:  $157.50 $121.50 
AMS Member Price:  $140.00 $108.00 

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 4manifolds. The second part investigates the main invariant of a 4manifold—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 differentialgeometric method has brought to light the unwieldy nature of smooth 4manifolds; 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 lowdimensional 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. Blowdown 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. Fibersums

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: selfduality

9.4. The Donaldson invariants

9.5. Notes

Chapter 10: The Seiberg–Witten Invariants

10.1. Almostcomplex 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 4manifolds, 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 4manifolds.
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 uptodate collection of references for the reader wishing to get a more detailed or rigorous knowledge of a specific topic. The exposition is userfriendly, with a large number of illustrations and examples.
Mathematical Reviews


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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 4manifolds. The second part investigates the main invariant of a 4manifold—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 differentialgeometric method has brought to light the unwieldy nature of smooth 4manifolds; 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 lowdimensional 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. Blowdown 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. Fibersums

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: selfduality

9.4. The Donaldson invariants

9.5. Notes

Chapter 10: The Seiberg–Witten Invariants

10.1. Almostcomplex 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 4manifolds, 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 4manifolds.
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 uptodate collection of references for the reader wishing to get a more detailed or rigorous knowledge of a specific topic. The exposition is userfriendly, with a large number of illustrations and examples.
Mathematical Reviews