##
**Instantons and four-manifolds.**
*(English)*
Zbl 0559.57001

Mathematical Sciences Research Institute Publications, Vol. 1. New York etc.: Springer-Verlag. X, 232 p. DM 48.00; $ 17.50 (1984).

This book, the first in the new Springer Series ”Publications of the Mathematical Sciences Research Institute” (in Berkeley) is the polished version of notes of a seminar held at this institution in 1982-1983.

It provides a proof with comments of S. Donaldson’s theorem to the effect that, ”if definite, the intersection form of a simply connected oriented 4-manifold is standard (i.e., that the manifold is oriented cobordant to a connected sum of complex projective planes)”. Combined with some topological arguments mainly due to M. Freedman, this theorem leads to the existence of non standard differentiable structures on the topological space \({\mathbb{R}}^ 4.\)

If one looks back at how the various ingredients of the proof came about, this spectacular outcome can be viewed as an exceptional pay-off for the recent interest of mathematicians in Yang-Mills theory, a variational problem of great importance to theoretical physicists (mainly because it provides classical models for some quantum phenomena).

The 5-manifold realizing the sought for cobordism is the moduli space \({\mathcal M}\) of minimizing solutions of this variational problem when properly set on the space of connections over the simplest non-trivial \(SU_ 2\)-bundle over the manifold. The local structure of this space (if non empty) was well understood since the work of Atiyah-Hitchin-Singer. Donaldson’s main contribution has been to give a complete description of the global structure of \({\mathcal M}\), a crucial step in this approach being C. Taubes’ existence theorem for self-dual connections together with some regularity and compactness theorems due to K. Uhlenbeck.

At least, three areas of mathematics (topology, differential geometry, and analysis) interact in the proof. These notes intend to introduce the reader to all of them. A quick look at the Table of Contents (see below) shows that great efforts in this direction have been made.

We reproduce here the Table of Contents: Introduction; Glossary; § 1. Fake \({\mathbb{R}}^ 4\); § 2. The Yang-Mills equations; § 3. Manifolds of connections; § 4. Cones on \({\mathbb{C}}P^ 2\); § 5. Orientability; § 6. Introduction to Taubes’ Theorem; § 7. Taubes’ Theorem; § 8. Compactness; § 9. The Collar theorem; § 10. The technique of Fintushel and Stern; Ap. A. The group of Sobolev gauge transformations; Ap. B. The Pontryagin-Thom construction; Ap. C. Weitzenböck formulas; Ap. D. The removability of singularities; Ap. E. Topological remarks.

Here is a more detailed review of the contents of the book. It opens with a welcome introduction providing some perspective on the impact of the result presented, and an overview on the contents of each chapter.

Chapters 1, 5, and 10 together with Appendices B and E present the necessary topological ingredients. In Chapter 1, it is sketched how Donaldson’s result leads to the existence of non standard differentiable structures over \({\mathbb{R}}^ 4\). Orientability of the moduli space \({\mathcal M}\) requires some discussion of homotopy classes of maps into \(S^ 3\) and topological properties of the action of the gauge group (Chapter 5 and Appendix B). General facts of interest on the intersection form and on the classification of bundles are collected in Appendix E.

Chapter 10 describes results of Fintushel and Stern who, by using \(SO_ 3\) rather than \(SU_ 2\)-bundles and less analysis, have also been able to exclude a number of integral quadratic forms as intersection forms of differentiable oriented 4-manifolds with finite fundamental group.

The differential geometric set-up is described along Chapters 2, 4, and Appendix C, and parts of Chapter 6. Chapter 2 introduces the Yang-Mills functional and Yang-Mills equations. A local description of the moduli space \({\mathcal M}\) near singular points is provided in Chapter 4 by a discussion of the fundamental elliptic complex. Appendix C discusses Weitzenböck formulae relating Laplacians on vector-valued forms to the so-called rough Laplacians. These are basic in obtaining vanishing theorems via the Bochner trick. The first half of Chapter 6 presents the basic family of instanton solutions of Yang-Mills equations over \(S^ 4\) via quaternionic calculus which serve as universal models all along the proof.

The longest developments are devoted to the analytic side of the proof. It is indeed the most delicate part, and requires the use of elaborate techniques in non-linear PDE theory; which presently are not easily accessible in book form. Chapter 3 (together with the second half of Chapter 6) contains the functional analytic set-up for the weak formulation of Yang-Mills equations. This includes a slice theorem, basic for the study of the action of the gauge group on the space of connections. (This is taken up again in Appendix A.)

A simplified proof of Taubes’ existence theorem is described in Chapter 7. The new idea is to obtain the self-dual connection by the continuity method after the metric of the base manifold has been blown up at the center of the ball over which the standard instanton solution over \(S^ 4\) has been grafted. This is possible because of the conformal invariance in dimension 4 of the Yang-Mills functional. (The same idea is also used in Appendix D to simplify K. Uhlenbeck’s removability of point singularities theorem.) Conformal invariance of the equations is also connected with the lack of compactness of the space of solutions, and to the occurence of limiting Sobolev exponents in the weak formulation of the problem. This is the theme of Chapter 8 in which non compact parts of the moduli space \({\mathcal M}\) are studied. The analytical arguments used there are presented at length. This leads in Chapter 9 to the collar theorem describing a neighbourhood of infinity in \({\mathcal M}\) as a product of the base manifold by an open interval.

The book can certainly be used at various levels, either as an introduction to gauge theory (and also to non-linear PDEs of geometric interest), or to help organizing a seminar on Donaldson’s theorem and on subsequent spectacular applications of gauge theory to topology. Efforts have clearly been made to ease access to the book by many potential readers. They should be successful. The word processor generated a readable, if not very beautiful, text. (Italics in particular look quite ugly.) The drawings could have been made more carefully, but they are not misleading. There seem to be only few misprints. (The only serious one caught by the reviewer is a reversed inequality in the proof of Lemma D.2 on page 215.)

All in all, a good start for this new series!

It provides a proof with comments of S. Donaldson’s theorem to the effect that, ”if definite, the intersection form of a simply connected oriented 4-manifold is standard (i.e., that the manifold is oriented cobordant to a connected sum of complex projective planes)”. Combined with some topological arguments mainly due to M. Freedman, this theorem leads to the existence of non standard differentiable structures on the topological space \({\mathbb{R}}^ 4.\)

If one looks back at how the various ingredients of the proof came about, this spectacular outcome can be viewed as an exceptional pay-off for the recent interest of mathematicians in Yang-Mills theory, a variational problem of great importance to theoretical physicists (mainly because it provides classical models for some quantum phenomena).

The 5-manifold realizing the sought for cobordism is the moduli space \({\mathcal M}\) of minimizing solutions of this variational problem when properly set on the space of connections over the simplest non-trivial \(SU_ 2\)-bundle over the manifold. The local structure of this space (if non empty) was well understood since the work of Atiyah-Hitchin-Singer. Donaldson’s main contribution has been to give a complete description of the global structure of \({\mathcal M}\), a crucial step in this approach being C. Taubes’ existence theorem for self-dual connections together with some regularity and compactness theorems due to K. Uhlenbeck.

At least, three areas of mathematics (topology, differential geometry, and analysis) interact in the proof. These notes intend to introduce the reader to all of them. A quick look at the Table of Contents (see below) shows that great efforts in this direction have been made.

We reproduce here the Table of Contents: Introduction; Glossary; § 1. Fake \({\mathbb{R}}^ 4\); § 2. The Yang-Mills equations; § 3. Manifolds of connections; § 4. Cones on \({\mathbb{C}}P^ 2\); § 5. Orientability; § 6. Introduction to Taubes’ Theorem; § 7. Taubes’ Theorem; § 8. Compactness; § 9. The Collar theorem; § 10. The technique of Fintushel and Stern; Ap. A. The group of Sobolev gauge transformations; Ap. B. The Pontryagin-Thom construction; Ap. C. Weitzenböck formulas; Ap. D. The removability of singularities; Ap. E. Topological remarks.

Here is a more detailed review of the contents of the book. It opens with a welcome introduction providing some perspective on the impact of the result presented, and an overview on the contents of each chapter.

Chapters 1, 5, and 10 together with Appendices B and E present the necessary topological ingredients. In Chapter 1, it is sketched how Donaldson’s result leads to the existence of non standard differentiable structures over \({\mathbb{R}}^ 4\). Orientability of the moduli space \({\mathcal M}\) requires some discussion of homotopy classes of maps into \(S^ 3\) and topological properties of the action of the gauge group (Chapter 5 and Appendix B). General facts of interest on the intersection form and on the classification of bundles are collected in Appendix E.

Chapter 10 describes results of Fintushel and Stern who, by using \(SO_ 3\) rather than \(SU_ 2\)-bundles and less analysis, have also been able to exclude a number of integral quadratic forms as intersection forms of differentiable oriented 4-manifolds with finite fundamental group.

The differential geometric set-up is described along Chapters 2, 4, and Appendix C, and parts of Chapter 6. Chapter 2 introduces the Yang-Mills functional and Yang-Mills equations. A local description of the moduli space \({\mathcal M}\) near singular points is provided in Chapter 4 by a discussion of the fundamental elliptic complex. Appendix C discusses Weitzenböck formulae relating Laplacians on vector-valued forms to the so-called rough Laplacians. These are basic in obtaining vanishing theorems via the Bochner trick. The first half of Chapter 6 presents the basic family of instanton solutions of Yang-Mills equations over \(S^ 4\) via quaternionic calculus which serve as universal models all along the proof.

The longest developments are devoted to the analytic side of the proof. It is indeed the most delicate part, and requires the use of elaborate techniques in non-linear PDE theory; which presently are not easily accessible in book form. Chapter 3 (together with the second half of Chapter 6) contains the functional analytic set-up for the weak formulation of Yang-Mills equations. This includes a slice theorem, basic for the study of the action of the gauge group on the space of connections. (This is taken up again in Appendix A.)

A simplified proof of Taubes’ existence theorem is described in Chapter 7. The new idea is to obtain the self-dual connection by the continuity method after the metric of the base manifold has been blown up at the center of the ball over which the standard instanton solution over \(S^ 4\) has been grafted. This is possible because of the conformal invariance in dimension 4 of the Yang-Mills functional. (The same idea is also used in Appendix D to simplify K. Uhlenbeck’s removability of point singularities theorem.) Conformal invariance of the equations is also connected with the lack of compactness of the space of solutions, and to the occurence of limiting Sobolev exponents in the weak formulation of the problem. This is the theme of Chapter 8 in which non compact parts of the moduli space \({\mathcal M}\) are studied. The analytical arguments used there are presented at length. This leads in Chapter 9 to the collar theorem describing a neighbourhood of infinity in \({\mathcal M}\) as a product of the base manifold by an open interval.

The book can certainly be used at various levels, either as an introduction to gauge theory (and also to non-linear PDEs of geometric interest), or to help organizing a seminar on Donaldson’s theorem and on subsequent spectacular applications of gauge theory to topology. Efforts have clearly been made to ease access to the book by many potential readers. They should be successful. The word processor generated a readable, if not very beautiful, text. (Italics in particular look quite ugly.) The drawings could have been made more carefully, but they are not misleading. There seem to be only few misprints. (The only serious one caught by the reviewer is a reversed inequality in the proof of Lemma D.2 on page 215.)

All in all, a good start for this new series!

Reviewer: J.-P.Bourguignon

### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C05 | Connections (general theory) |

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

57R19 | Algebraic topology on manifolds and differential topology |

11E16 | General binary quadratic forms |

58J10 | Differential complexes |

58J20 | Index theory and related fixed-point theorems on manifolds |

58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |

58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |

55R10 | Fiber bundles in algebraic topology |

55Q45 | Stable homotopy of spheres |