Lectures on Differential Topology

This book gives a comprehensive introduction to the theory of smooth manifolds, maps, and fundamental associated structures with an emphasis on “bare hands” approaches, combining differential-topological cut-and-paste procedures and applications of transversality. In particular, the smooth cobordism cup-product is defined from scratch and used as the main tool in a variety of settings. After establishing the fundamentals, the book proceeds to a broad range of more advanced topics in differential topology, including degree theory, the Poincaré-Hopf index theorem, bordism-characteristic numbers, and the Pontryagin-Thom construction. Cobordism intersection forms are used to classify compact surfaces; their quadratic enhancements are developed and applied to studying the homotopy groups of spheres, the bordism group of immersed surfaces in a 3-manifold, and congruences mod 16 for the signature of intersection forms of 4-manifolds. Other topics include the high-dimensional $h$-cobordism theorem stressing the role of the “Whitney trick”, a determination of the singleton bordism modules in low dimensions, and proofs of parallelizability of orientable 3-manifolds and the Lickorish-Wallace theorem. Nash manifolds and Nash's questions on the existence of real algebraic models are also discussed. This book will be useful as a textbook for beginning masters and doctoral students interested in differential topology, who have finished a standard undergraduate mathematics curriculum. It emphasizes an active learning approach, and exercises are included within the text as part of the flow of ideas. Experienced readers may use this book as a source of alternative, constructive approaches to results commonly presented in more advanced contexts with specialized techniques. Dedication 6 Contents 8 Preface 14 Introduction 16 1. The smooth category of open subsets of Euclidean spaces 32 2. The category of embedded smooth manifolds 62 3. Stiefel and Grassmann manifolds 74 4. The category of smooth manifolds 88 5. Tautological bundles and pull-back 120 6. Compact embedded smooth manifolds 138 7. Cut and paste compact manifolds 164 8. Transversality 194 9. Morse functions and handle decompositions 208 10. Bordism 222 11. Smooth cobordism 238 12. Applications of cobordism rings 250 13. Line bundles, hypersurfaces, and cobordism 262 14. Euler-Poincar ́e characteristic 272 15. Surfaces 286 16. Bordism characteristic numbers 308 17. The Pontryagin-Thom construction 318 18. High-dimensional manifolds 338 19. On 3-manifolds 348 20. On 4-manifolds 412 Appendix: Baby categories 444 Bibliography 448 Index 454 "The pdf contains a draft title page, draft copyright page and a draft manuscript"-- Provided by publisher