Lectures on differential topology
著者
書誌事項
Lectures on differential topology
（Graduate studies in mathematics, 218）
American Mathematical Society, c2021
- : hardcover
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注記
Includes bibliographical references (p. 417-422) and index
内容説明・目次
内容説明
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 Poincare-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.
目次
The smooth category of open subsets of Euclidean spaces
The category of embedded smooth manifolds
Stiefel and Grassmann manifolds
The category of smooth manifolds
Tautological bundles and pull-back
Compact embedded smooth manifolds
Cut and paste compact manifolds
Transversality
Morse functions and handle decompositions
Bordism
Smooth cobordism
Applications of cobordism rings
Line bundles, hypersurfaces, and cobordism
Euler-Poincare characteristic
Surfaces
Bordism characteristic numbers
The Pontryagin-Thom construction
High-dimensional manifolds
On 3-manifolds
On 4-manifolds
Baby categories
Bibliography
Index
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