Instantons and Four-ManifoldsFrom the reviews of the first edition: "This book exposes the beautiful confluence of deep techniques and ideas from mathematical physics and the topological study of the differentiable structure of compact four-dimensional manifolds, compact spaces locally modeled on the world in which we live and operate... The book is filled with insightful remarks, proofs, and contributions that have never before appeared in print. For anyone attempting to understand the work of Donaldson and the applications of gauge theories to four-dimensional topology, the book is a must." #Science#1 "I would strongly advise the graduate student or working mathematician who wishes to learn the analytic aspects of this subject to begin with Freed and Uhlenbeck's book." #Bulletin of the American Mathematical Society#2 |
Contents
Glossary | 12 |
Differentiable 4manifolds | 20 |
The YangMills Equations | 28 |
The YangMills functional | 34 |
Cones on | 63 |
5 Orientability | 74 |
6 Introduction to Taubes Theorem | 83 |
7 Taubes Theorem | 100 |
8 Compactness | 117 |
9 The Collar Theorem | 133 |
10 The Technique of Fintushel and Stern | 154 |
Appendix A The Group of Sobolev Gauge | 160 |
Weitzenböck Formulas | 169 |
The Removability of Singularities | 175 |
| 184 | |
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Common terms and phrases
2-forms 4-manifold action adjoint anti-self-dual argument b₁ Chern Chern class classifying cohomology compact compute concentrated instanton conformally invariant construction convergent coordinates COROLLARY curvature defined denote diffeomorphism differentiable structure dimension Donaldson's Theorem elliptic complex elliptic operators embedded equivariant estimate exists finite dimensional fixed follows framed cobordism Fredholm gauge transformations geometry H₁(M H²(M Hence Hilbert space homeomorphic homology homotopy implies instanton intersection form invertible kernel LEMMA Lie algebra line bundle linear manifold metric g moduli space nonlinear nontrivial norm obtain open set operator oriented orthogonal positive definite principal bundle proof PROPOSITION prove reducible connections result Riemannian satisfies self-dual connections simply connected smoothable Sobolev solutions Taubes theory topological trivial unique vanishes vector bundle Weitzenböck formula Yang-Mills connection Yang-Mills equations Ω²