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Lectures on Differential Topology

Lectures on Differential Topology PDF Author: Riccardo Benedetti
Publisher: American Mathematical Soc.
ISBN: 1470462710
Category : Education
Languages : en
Pages : 425

Book Description
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 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.

Lectures on Differential Topology

Lectures on Differential Topology PDF Author: Riccardo Benedetti
Publisher: American Mathematical Soc.
ISBN: 1470462710
Category : Education
Languages : en
Pages : 425

Book Description
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 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.

Lectures on Differential Geometry

Lectures on Differential Geometry PDF Author: Shlomo Sternberg
Publisher: American Mathematical Soc.
ISBN: 0821813854
Category : Mathematics
Languages : en
Pages : 466

Book Description
This book is based on lectures given at Harvard University during the academic year 1960-1961. The presentation assumes knowledge of the elements of modern algebra (groups, vector spaces, etc.) and point-set topology and some elementary analysis. Rather than giving all the basic information or touching upon every topic in the field, this work treats various selected topics in differential geometry. The author concisely addresses standard material and spreads exercises throughout the text. His reprint has two additions to the original volume: a paper written jointly with V. Guillemin at the beginning of a period of intense interest in the equivalence problem and a short description from the author on results in the field that occurred between the first and the second printings.

Introduction to Differential Topology

Introduction to Differential Topology PDF Author: Theodor Bröcker
Publisher: Cambridge University Press
ISBN: 9780521284707
Category : Mathematics
Languages : en
Pages : 176

Book Description
This book is intended as an elementary introduction to differential manifolds. The authors concentrate on the intuitive geometric aspects and explain not only the basic properties but also teach how to do the basic geometrical constructions. An integral part of the work are the many diagrams which illustrate the proofs. The text is liberally supplied with exercises and will be welcomed by students with some basic knowledge of analysis and topology.

Topology from the Differentiable Viewpoint

Topology from the Differentiable Viewpoint PDF Author: John Willard Milnor
Publisher: Princeton University Press
ISBN: 9780691048338
Category : Mathematics
Languages : en
Pages : 80

Book Description
This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.

Differential Topology

Differential Topology PDF Author: Amiya Mukherjee
Publisher: Birkhäuser
ISBN: 3319190458
Category : Mathematics
Languages : en
Pages : 357

Book Description
This book presents a systematic and comprehensive account of the theory of differentiable manifolds and provides the necessary background for the use of fundamental differential topology tools. The text includes, in particular, the earlier works of Stephen Smale, for which he was awarded the Fields Medal. Explicitly, the topics covered are Thom transversality, Morse theory, theory of handle presentation, h-cobordism theorem and the generalised Poincaré conjecture. The material is the outcome of lectures and seminars on various aspects of differentiable manifolds and differential topology given over the years at the Indian Statistical Institute in Calcutta, and at other universities throughout India. The book will appeal to graduate students and researchers interested in these topics. An elementary knowledge of linear algebra, general topology, multivariate calculus, analysis and algebraic topology is recommended.

Lectures on Algebraic and Differential Topology

Lectures on Algebraic and Differential Topology PDF Author: R. Bott
Publisher: Springer
ISBN: 354037616X
Category : Mathematics
Languages : en
Pages : 183

Book Description


Lectures on Differential Topology

Lectures on Differential Topology PDF Author: Riccardo Benedetti
Publisher:
ISBN: 9781470466732
Category : Differential topology
Languages : en
Pages : 425

Book Description
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 number.

Lectures on the Topology of 3-manifolds

Lectures on the Topology of 3-manifolds PDF Author: Nikolai Saveliev
Publisher: Walter de Gruyter
ISBN: 9783110162721
Category : Mathematics
Languages : en
Pages : 220

Book Description


Lectures on Algebraic and Differential Topology

Lectures on Algebraic and Differential Topology PDF Author: R. Bott
Publisher:
ISBN: 9783662178119
Category :
Languages : en
Pages : 188

Book Description


Calculus on Manifolds

Calculus on Manifolds PDF Author: Michael Spivak
Publisher: Westview Press
ISBN: 9780805390216
Category : Science
Languages : en
Pages : 164

Book Description
This book uses elementary versions of modern methods found in sophisticated mathematics to discuss portions of "advanced calculus" in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level.