2005;
614 pp;
Hardcover

MSC: Primary 57;
Secondary 14; 32

**Print ISBN: 978-0-8218-3749-8
Product Code: FOURMAN**

List Price: $80.00

AMS Member Price: $64.00

MAA Member Price: $72.00

#### Supplemental Materials

# The Wild World of 4-Manifolds

Share this page
*Alexandru Scorpan*

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.

#### Readership

Graduate students and research mathematicians interested in low-dimensional topology.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## The Wild World of 4-Manifolds

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents xi12
- Preview vii8 free
- Contents of the Notes xv16 free
- Introduction 122 free
- Front matter 526 free
- Part I: Background Scenery 2344
- Part II: Smooth 4–Manifolds and Intersection Forms 105126
- Part III: A Survey of Complex Surfaces 271292
- Part IV: Gauge Theory on 4–Manifolds 325346
- Epilogue 557578
- List of Figures and Tables 559580
- Bibliography 567588
- Index 587608
- Errata 611632
- Back Cover Back Cover1636