Lectures On Differential Topology

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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.

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Genre : Education
Author : Riccardo Benedetti
Publisher : American Mathematical Soc.
Release : 2021-10-27
File : 425 Pages
ISBN-13 : 9781470466749


Lectures On Algebraic And Differential Topology

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Genre : Mathematics
Author : R. Bott
Publisher : Springer
Release : 2006-11-15
File : 183 Pages
ISBN-13 : 9783540376163


Lectures On Differential Geometry

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This book presents a concise introduction to differential geometry. It is aimed at advanced undergraduate students and first year graduate students who wish to have a basic solid knowledge of the subject, and it can serve as a starting point for more advanced reading. The book is organized into lectures, so it can easily be used as a textbook for a beginning graduate-level course in differential geometry.

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Genre : Mathematics
Author : Rui Loja Fernandes
Publisher : World Scientific
Release : 2024-09-03
File : 419 Pages
ISBN-13 : 9789811252662


Differential Topology

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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.

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Genre : Mathematics
Author : Amiya Mukherjee
Publisher : Birkhäuser
Release : 2015-06-30
File : 357 Pages
ISBN-13 : 9783319190457


Differential Topology

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A. Banyaga: On the group of diffeomorphisms preserving an exact symplectic.- G.A. Fredricks: Some remarks on Cauchy-Riemann structures.- A. Haefliger: Differentiable Cohomology.- J.N. Mather: On the homology of Haefliger’s classifying space.- P. Michor: Manifolds of differentiable maps.- V. Poenaru: Some remarks on low-dimensional topology and immersion theory.- F. Sergeraert: La classe de cobordisme des feuilletages de Reeb de S3 est nulle.- G. Wallet: Invariant de Godbillon-Vey et difféomorphismes commutants.

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Genre : Mathematics
Author : V. Villani
Publisher : Springer Science & Business Media
Release : 2011-06-07
File : 149 Pages
ISBN-13 : 9783642111020


Differential Topology Differential Geometry And Applications

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Genre : Differential topology
Author : Shiing-Shen Chern
Publisher :
Release : 1972
File : 344 Pages
ISBN-13 : STANFORD:36105031579522


Differential Topology

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...there are reasons enough to warrant a coherent treatment of the main body of differential topology in the realm of Banach manifolds, which is at the same time correct and complete. This book fills the gap: whenever possible the manifolds treated are Banach manifolds with corners. Corners add to the complications and the authors have carefully fathomed the validity of all main results at corners. Even in finite dimensions some results at corners are more complete and better thought out here than elsewhere in the literature. The proofs are correct and with all details. I see this book as a reliable monograph of a well-defined subject; the possibility to fall back to it adds to the feeling of security when climbing in the more dangerous realms of infinite dimensional differential geometry. Peter W. Michor

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Genre : Mathematics
Author : J. Margalef-Roig
Publisher : Elsevier
Release : 1992-06-02
File : 622 Pages
ISBN-13 : 9780444884343


Elementary Differential Topology Am 54 Volume 54

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The description for this book, Elementary Differential Topology. (AM-54), Volume 54, will be forthcoming.

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Genre : Mathematics
Author : James R. Munkres
Publisher : Princeton University Press
Release : 2016-03-02
File : 112 Pages
ISBN-13 : 9781400882656


Lectures On Field Theory And Topology

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These lectures recount an application of stable homotopy theory to a concrete problem in low energy physics: the classification of special phases of matter. While the joint work of the author and Michael Hopkins is a focal point, a general geometric frame of reference on quantum field theory is emphasized. Early lectures describe the geometric axiom systems introduced by Graeme Segal and Michael Atiyah in the late 1980s, as well as subsequent extensions. This material provides an entry point for mathematicians to delve into quantum field theory. Classification theorems in low dimensions are proved to illustrate the framework. The later lectures turn to more specialized topics in field theory, including the relationship between invertible field theories and stable homotopy theory, extended unitarity, anomalies, and relativistic free fermion systems. The accompanying mathematical explanations touch upon (higher) category theory, duals to the sphere spectrum, equivariant spectra, differential cohomology, and Dirac operators. The outcome of computations made using the Adams spectral sequence is presented and compared to results in the condensed matter literature obtained by very different means. The general perspectives and specific applications fuse into a compelling story at the interface of contemporary mathematics and theoretical physics.

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Genre : Mathematics
Author : Daniel S. Freed
Publisher : American Mathematical Soc.
Release : 2019-08-23
File : 202 Pages
ISBN-13 : 9781470452063


Differential Topology

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Exploring the full scope of differential topology, this comprehensive account of geometric techniques for studying the topology of smooth manifolds offers a wide perspective on the field. Building up from first principles, concepts of manifolds are introduced, supplemented by thorough appendices giving background on topology and homotopy theory. Deep results are then developed from these foundations through in-depth treatments of the notions of general position and transversality, proper actions of Lie groups, handles (up to the h-cobordism theorem), immersions and embeddings, concluding with the surgery procedure and cobordism theory. Fully illustrated and rigorous in its approach, little prior knowledge is assumed, and yet growing complexity is instilled throughout. This structure gives advanced students and researchers an accessible route into the wide-ranging field of differential topology.

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Genre : Mathematics
Author : C. T. C. Wall
Publisher : Cambridge University Press
Release : 2016-07-04
File : 355 Pages
ISBN-13 : 9781107153523