Sinposis

THE CONCEPTS OF DIFFERENTIAL TOPOLOGY FORM THE CENTER OF MANY MATHEMATICAL DISCIPLINES SUCH AS DIFFERENTIAL GEOMETRY AND LIE GROUP THEORY. DIFFERENTIAL MANIFOLDS PRESENTS TO ADVANCED UNDERGRADUATES AND GRADUATE STUDENTS THE SYSTEMATIC STUDY OF THE TOPOLOGICAL STRUCTURE OF SMOOTH MANIFOLDS. AUTHOR ANTONI A. KOSINSKI, PROFESSOR EMERITUS OF MATHEMATICS AT RUTGERS UNIVERSITY, OFFERS AN ACCESSIBLE APPROACH TO BOTH THE H-COBORDISM THEOREM AND THE CLASSIFICATION OF DIFFERENTIAL STRUCTURES ON SPHERES. "HOW USEFUL IT IS," NOTED THE BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, "TO HAVE A SINGLE, SHORT, WELL-WRITTEN BOOK ON DIFFERENTIAL TOPOLOGY." THIS VOLUME BEGINS WITH A DETAILED, SELF-CONTAINED REVIEW OF THE FOUNDATIONS OF DIFFERENTIAL TOPOLOGY THAT REQUIRES ONLY A MINIMAL KNOWLEDGE OF ELEMENTARY ALGEBRAIC TOPOLOGY. SUBSEQUENT CHAPTERS EXPLAIN THE TECHNIQUE OF JOINING MANIFOLDS ALONG SUBMANIFOLDS, THE HANDLE PRESENTATION THEOREM, AND THE PROOF OF THE H-COBORDISM THEOREM BASED ON THESE CONSTRUCTIONS. THERE FOLLOWS A CHAPTER ON THE PONTRIAGIN CONSTRUCTION_THE PRINCIPAL LINK BETWEEN DIFFERENTIAL TOPOLOGY AND HOMOTOPY THEORY. THE FINAL CHAPTER INTRODUCES THE METHOD OF SURGERY AND APPLIES IT TO THE CLASSIFICATION OF SMOOTH STRUCTURES OF SPHERES. THE TEXT IS SUPPLEMENTED BY NUMEROUS INTERESTING HISTORICAL NOTES AND CONTAINS A NEW APPENDIX, "THE WORK OF GRIGORY PERELMAN," BY JOHN W. MORGAN, WHICH DISCUSSES THE MOST RECENT DEVELOPMENTS IN DIFFERENTIAL TOPOLOGY.