##
**The wild world of 4-manifolds.**
*(English)*
Zbl 1075.57001

Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3749-4/hbk). xv, 609 p. (2005).

The title of this book refers to the now well-known facts that differential topology in dimension 4 is very different from that in all other dimensions, and that there is as yet no comprehensive organization of the observed phenomena. Since 1990 there have been a number of books on aspects of 4-manifold theory: the foundational book on the topology of 4-manifolds by [M. H. Freedman and F. S. Quinn, Topology of 4-manifolds (1990; Zbl 0705.57001)], half a dozen books on gauge theoretic topology [S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds (1990; Zbl 0820.57002); M. Marcolli, Seiberg-Witten gauge theory (1999; Zbl 0954.53003); W. J. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds (1996; Zbl 0846.57001); J. D. Moore, Lectures on Seiberg-Witten invariants (1996; Zbl 0857.58001); L. I. Nicolaescu, Notes on Seiberg-Witten theory (2000; Zbl 0978.57027); T. Petrie and J. Randall, Connections, definite forms, and four-manifolds (1990; Zbl 0725.53001)], two on the differential topology of complex surfaces [R. Friedman and J. W. Morgan, Smooth four-manifolds and complex surfaces (1994; Zbl 0817.14017); J. W. Morgan and O’Grady, Differential topology of complex surfaces. Elliptic surfaces with \(p_ g=1\): smooth classification (1993; Zbl 0789.14037)], one on the Kirby calculus [R. E. Gompf and A. I. Stipsicz, 4-manifolds and Kirby calculus (1999; Zbl 0933.57020)], one on symplectic and contact topology [B. Ozbagci and A. I. Stipsicz, Surgery on contact 3-manifolds and Stein surfaces (2004; Zbl 1067.57024)], the reviewer’s monograph on the algebraic topology of non-simply connected \(PD_4\)-complexes and geometric 4-manifolds [J. Hillman, Four-manifolds, geometries and knots (2002; Zbl 1087.57015)], H. J. Baues’ work on the combinatorial homotopy of 4-dimensional complexes [Combinatorial homotopy and 4-dimensional complexes (1991; Zbl 0716.55001)] and the Borel seminar on fundamental groups of compact Kähler manifolds [J. Amorós, M. Burger, A. Corlette, D. Kotschick and D. Toledo, Fundamental groups of compact Kähler manifolds (1996; Zbl 0849.32006)]. Before these were the lecture notes of R. C. Kirby [The topology of 4-manifolds (1989; Zbl 0668.57001)] and the BAMS survey article by R. Mandelbaum [Bull. Am. Math. Soc., New Ser. 2, 1–159 (1980; Zbl 0476.57005)], written several years before the major work of Freedman and Donaldson.

The present book is closest in spirit to the latter two items, and indeed most of the first two thirds of the book could have been written twenty years ago. (The final third considers much more recent work). It is neither a text nor an encyclopedic reference. Instead it is a survey of the realm of 1-connected smooth 4-manifolds. Each chapter is followed by extensive notes and guides to further reading, which together occupy almost half the text.

The first Part provides the broader context, summarizing fundamental elements of the classification of manifolds in higher (and lower) dimensions. Chapter 1 considers handle decompositions, the \(s\)-cobordism theorem and the Whitney trick. However surgery is scarcely mentioned (and is not in the index), while smoothing theory is deferred to the notes for Chapter 4. Chapter 2 considers the topology of 4-manifolds: Casson handles, plumbing, homology 3-spheres bound contractible 4-manifolds, the 4-dimensional topological \(h\)-cobordism theorem, a brief discussion of the Kirby calculus and two pages on 3-manifolds.

Part II (Chapters 3-5) considers the intersection form, which is the most important algebraic invariant of a closed 4-manifold. In particular, Chapter 4 includes the theorems of Whitehead and Wall on the determination of 1-connected smooth 4-manifolds up to \(h\)-cobordism by their intersection forms and Rochlin’s parity constraint on the signature of Spin 4-manifolds. Chapter 5 states Freedman’s topological classification of 1-connected 4-manifolds and Donaldson’s diagonalization theorem for definite intersection forms, and derives examples of exotic smooth structures on \(\mathbb R^4\). The notes for this part consider Spin structures, Čech cocycles for characteristic classes, obstruction theory, smoothing theory and the Pontrjagin-Thom construction. Part III (Chapters 6-8) is a quick survey of the classification of complex surfaces, with extended discussions of “the” K3 surface and of elliptic surfaces. Symplectic structures and the adjunction formula are introduced here. This is the shortest section of the book, and the notes for these chapters are brief.

The final Part (which is also the longest part) is devoted to gauge theory. Chapter 9 gives a short outline of Donaldson invariants. There is then a longer chapter on the Seiberg-Witten equations, and two final chapters on applications of SW theory. Chapter 11 considers the minimal genus problem for surfaces representing homology classes in symplectic manifolds, the adjunction inequality now playing a key role. Chapter 12 outlines the work of R. Fintushel and R. J. Stern on modifying the smooth structure of a 4-manifold by “knot surgery” [Invent. Math. 134, No. 2, 363–400 (1998; Zbl 0914.57015)]. (The treatment of the Alexander polynomial on page 540 is oversimplified; in general the knot module is not a cyclic \(\mathbb{Z}[t,t^{-1}]\)-module.) The notes for this section include Lefshetz fibrations, an SW proof of Donaldson’s diagonalization theorem, the Gromov-Taubes invariants of symplectic 4-manifolds, the Arf invariant, the characteristic cobordism group and its applications to extensions of Rochlin’s Theorem, rational blowdowns and more. The bibliography is detailed and up to date, with fastidious attention to distinctions between successive editions. (References [FM94a] and [FM94b] appear to be identical!)

I believe the author has achieved his goal of writing a book that can be read with profit by readers of varying backgrounds. As someone whose interest in manifolds began in the era of surgery, I regret the rigorous exclusion of the fundamental group, but I found this an attractive book for browsing, and for conveying an overview of topics I would like to understand better.

The present book is closest in spirit to the latter two items, and indeed most of the first two thirds of the book could have been written twenty years ago. (The final third considers much more recent work). It is neither a text nor an encyclopedic reference. Instead it is a survey of the realm of 1-connected smooth 4-manifolds. Each chapter is followed by extensive notes and guides to further reading, which together occupy almost half the text.

The first Part provides the broader context, summarizing fundamental elements of the classification of manifolds in higher (and lower) dimensions. Chapter 1 considers handle decompositions, the \(s\)-cobordism theorem and the Whitney trick. However surgery is scarcely mentioned (and is not in the index), while smoothing theory is deferred to the notes for Chapter 4. Chapter 2 considers the topology of 4-manifolds: Casson handles, plumbing, homology 3-spheres bound contractible 4-manifolds, the 4-dimensional topological \(h\)-cobordism theorem, a brief discussion of the Kirby calculus and two pages on 3-manifolds.

Part II (Chapters 3-5) considers the intersection form, which is the most important algebraic invariant of a closed 4-manifold. In particular, Chapter 4 includes the theorems of Whitehead and Wall on the determination of 1-connected smooth 4-manifolds up to \(h\)-cobordism by their intersection forms and Rochlin’s parity constraint on the signature of Spin 4-manifolds. Chapter 5 states Freedman’s topological classification of 1-connected 4-manifolds and Donaldson’s diagonalization theorem for definite intersection forms, and derives examples of exotic smooth structures on \(\mathbb R^4\). The notes for this part consider Spin structures, Čech cocycles for characteristic classes, obstruction theory, smoothing theory and the Pontrjagin-Thom construction. Part III (Chapters 6-8) is a quick survey of the classification of complex surfaces, with extended discussions of “the” K3 surface and of elliptic surfaces. Symplectic structures and the adjunction formula are introduced here. This is the shortest section of the book, and the notes for these chapters are brief.

The final Part (which is also the longest part) is devoted to gauge theory. Chapter 9 gives a short outline of Donaldson invariants. There is then a longer chapter on the Seiberg-Witten equations, and two final chapters on applications of SW theory. Chapter 11 considers the minimal genus problem for surfaces representing homology classes in symplectic manifolds, the adjunction inequality now playing a key role. Chapter 12 outlines the work of R. Fintushel and R. J. Stern on modifying the smooth structure of a 4-manifold by “knot surgery” [Invent. Math. 134, No. 2, 363–400 (1998; Zbl 0914.57015)]. (The treatment of the Alexander polynomial on page 540 is oversimplified; in general the knot module is not a cyclic \(\mathbb{Z}[t,t^{-1}]\)-module.) The notes for this section include Lefshetz fibrations, an SW proof of Donaldson’s diagonalization theorem, the Gromov-Taubes invariants of symplectic 4-manifolds, the Arf invariant, the characteristic cobordism group and its applications to extensions of Rochlin’s Theorem, rational blowdowns and more. The bibliography is detailed and up to date, with fastidious attention to distinctions between successive editions. (References [FM94a] and [FM94b] appear to be identical!)

I believe the author has achieved his goal of writing a book that can be read with profit by readers of varying backgrounds. As someone whose interest in manifolds began in the era of surgery, I regret the rigorous exclusion of the fundamental group, but I found this an attractive book for browsing, and for conveying an overview of topics I would like to understand better.

Reviewer: Jonathan A. Hillman (Sydney)

### MSC:

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

57R57 | Applications of global analysis to structures on manifolds |

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |