Differential Topology

Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea--transversality--the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincaré-Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course.

Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea—transversality—the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincare-Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course.

Contents Preface Straight Forward to the Student Table of Symbols 1 Manifolds and Smooth Maps 1.1. Definitions 1.2. Derivatiuves and Tangents 1.3. The Inverse Function Theorem and Immersions 1.4. Submersions 1.5. Transversality 1.6. Homotopy and Stability 1.7. Sard's Theorem and Morse Functions 1.8. Embedding Manifolds in Euclidean Space 2 Transversality and Intersection 2.1. Manifolds with Boundary 2.2. One-Manifolds and Some Consequences 2.3. Transversality 2.4. Intersection Theory Mod 2 2.5. Winding Numbers and the Jordan-Brouwer Separation Theorem 2.6. The Borsuk-Ulam Theorem 3 Oriented Intersection Theory 3.1. Motivation 3.2. Orientation 3.3. Oriented Intersection Number 3.4. Lefschetz Fixed-Point Theory 3.5. Vector Fields and the Poincaré-Hopf Theorem 3.6. The Hopf Degree Theorem 3.7. The Euler Characteristic and Triangulations 4 Integration on Manifolds 4.1. Introduction 4.2. Exterior Algebra 4.3. Differential Forms 4.4. Integration on Manifolds 4.5. Exterior Derivative 4.5. Cohomology with Forms 4.7. Stokes Theorem 4.8. Integration and Mappings 4.9. The Gauss'Bonnet Theorem APPENDIX 1 Measure Zero and Sard's Theorem APPENDIX 2 Classification of Compact One-Manifolds Bibliography Index