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From 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
List of contents
to the First Edition.-
1 Fake ?4.- Differentiable structures.- Topological 4-manifolds.- Differentiable 4-manifolds.- A surgical failure.-
2 The Yang-Mills Equations.- Connections.- Topological quantum numbers.- The Yang-Mills functional.- Line bundles.- Donaldson's Theorem.-
3 Manifolds of Connections.- Sobolev spaces.- Reducible connections.- A slice theorem.- The parametrized moduli space.- The moduli space.-
4 Cones on ??2.- Slices again.- Structure of the singular point.- Perturbing the metric.-
5 Orientability.- Index bundles.- Components of.- The element - 1.-
6 Introduction to Taubes' Theorem.- Instantons on S4.- A grafting procedure.- Tools from analysis.- Analytic properties of SDYME.-
7 Taubes' Theorem.- Blowing up the metric.- The eigenvalue estimate.- The linearized equation.- Taubes' projection.-
8 Compactness.- Compactness and regularity.- Measuring concentrated curvature.- Compactness in M.-
9 The Collar Theorem.- Decay estimates.- Conformal deformations.- Exponential gauges.- Connectivity of the collar.-
10 The Technique of Fintushel and Stern.- The moduli space for SO(3) bundles.- Reducible connections.- Analytic details.- Appendix A The Group of Sobolev Gauge Transformations.- Appendix B The Pontrjagin-Thom Construction.- Appendix C Weitzenböck Formulas.- Appendix D The Removability of Singularities.- Appendix E Topological Remarks.
Report
From 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)
"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)
