Fiber Bundles And Homotopy
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Fiber Bundles And Homotopy
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Author : Dai Tamaki
language : en
Publisher: World Scientific
Release Date : 2021-05-17
Fiber Bundles And Homotopy written by Dai Tamaki and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-05-17 with Mathematics categories.
This book is an introduction to fiber bundles and fibrations. But the ultimate goal is to make the reader feel comfortable with basic ideas in homotopy theory. The author found that the classification of principal fiber bundles is an ideal motivation for this purpose. The notion of homotopy appears naturally in the classification. Basic tools in homotopy theory such as homotopy groups and their long exact sequence need to be introduced. Furthermore, the notion of fibrations, which is one of three important classes of maps in homotopy theory, can be obtained by extracting the most essential properties of fiber bundles. The book begins with elementary examples and then gradually introduces abstract definitions when necessary. The reader is assumed to be familiar with point-set topology, but it is the only requirement for this book.
Basic Topology 3
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Author : Mahima Ranjan Adhikari
language : en
Publisher: Springer Nature
Release Date : 2023-03-15
Basic Topology 3 written by Mahima Ranjan Adhikari and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-03-15 with Mathematics categories.
This third of the three-volume book is targeted as a basic course in algebraic topology and topology for fiber bundles for undergraduate and graduate students of mathematics. It focuses on many variants of topology and its applications in modern analysis, geometry, and algebra. Topics covered in this volume include homotopy theory, homology and cohomology theories, homotopy theory of fiber bundles, Euler characteristic, and the Betti number. It also includes certain classic problems such as the Jordan curve theorem along with the discussions on higher homotopy groups and establishes links between homotopy and homology theories, axiomatic approach to homology and cohomology as inaugurated by Eilenberg and Steenrod. It includes more material than is comfortably covered by beginner students in a one-semester course. Students of advanced courses will also find the book useful. This book will promote the scope, power and active learning of the subject, all the while covering a wide range of theory and applications in a balanced unified way.
Fibre Bundles
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Author : D. Husemöller
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
Fibre Bundles written by D. Husemöller and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-29 with Mathematics categories.
The notion of a fibre bundle first arose out of questions posed in the 1930s on the topology and geometry of manifolds. By the year 1950 the defini tion of fibre bundle had been clearly formulated, the homotopy classifica tion of fibre bundles achieved, and the theory of characteristic classes of fibre bundles developed by several mathematicians, Chern, Pontrjagin, Stiefel, and Whitney. Steenrod's book, which appeared in 1950, gave a coherent treatment of the subject up to that time. About 1955 Milnor gave a construction of a universal fibre bundle for any topological group. This construction is also included in Part I along with an elementary proof that the bundle is universal. During the five years from 1950 to 1955, Hirzebruch clarified the notion of characteristic class and used it to prove a general Riemann-Roch theorem for algebraic varieties. This was published in his Ergebnisse Monograph. A systematic development of characteristic classes and their applications to manifolds is given in Part III and is based on the approach of Hirze bruch as modified by Grothendieck.
Fibrations And Their Classification
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Author : Petar Pavešić
language : en
Publisher:
Release Date : 2013
Fibrations And Their Classification written by Petar Pavešić and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Fiber bundles (Mathematics) categories.
The concept of fibration is one of the great unifying mathematical ideas. It was initially introduced around 1930 in geometry and topology, and gradually expanded into many other parts of mathematics. Together with fibre bundles (which precedeed fibrations), they give formal expression to the idea of a continuous family of spaces, and of operations on such families. This monograph contains an exposition of the fundamental ideas of the theory of fibrations with particular emphasis on their classification. It deals at length with various types of fibrations as defined by Hurewicz, Dold and Serre, as well as the quasifibrations of Dold and Thom. The relationship between these concepts is analyzed in depth, with examples and counter-examples given. One of the salient properties of fibre bundles is that they are classified by homotopy classes of maps into some special spaces called classifying spaces. The classifying theory for fibrations is presented both abstractly, through the theory of representable functors, and constructively, by describing various models, like those introduced by Dold and Lashof, and by Milgram and Steenrod. In the couple of decades following their intoduction, the growth of the theory of fibrations resulted in a plethora of similar and interrelated theories and classification results for vector bundles, general fibre bundles, and other types of fibre spaces. As a new organizational principle, Peter May invented the concept of F-fibrations that generalizes all of the above, and is at the same time sufficiently structured to admit workable classification objects. The second part of the book is dedicated to an in-depth discussion of the theory of F-fibrations. The book is reasonably self-contained and the reader is assumed to have only some knowledge of general topology and basic homotopy theory, including elementary properties of homotopy groups. However, one must be aware that the level of exposition is at some places more advanced, and for these a prior course in algebraic topology or in the theory of fibre bundles would be very helpful, both as a motivation for the problems that are studied, as well as a measure of the required mathematical sophistication. The book can be used both as a text-book or as a reference. Most chapters are concluded with historical notes, tracing the origins of the concepts and the developments related to the classification of fibre bundles and fibrations.
Fibre Bundles
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Author : Dale Husemöller
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Fibre Bundles written by Dale Husemöller and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.
Basic properties, homotopy classification, and characteristic classes of fibre bundles have become an essential part of graduate mathematical education for students in geometry and mathematical physics. The new edition of this text includes two additional chapters, one on the gauge group of a bundle and the other on the differential forms representing characteristic classes of complex vector bundles on manifolds.
The Topology Of Fibre Bundles
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Author : Norman Steenrod
language : en
Publisher: Princeton University Press
Release Date : 1999-04-25
The Topology Of Fibre Bundles written by Norman Steenrod and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-04-25 with Mathematics categories.
Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory. This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject systematically. It begins with a general introduction to bundles, including such topics as differentiable manifolds and covering spaces. The author then provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles. The result is a classic and timeless work of great utility that will appeal to serious mathematicians and theoretical physicists alike.
Algebraic Topology From A Homotopical Viewpoint
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Author : Marcelo Aguilar
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-02-02
Algebraic Topology From A Homotopical Viewpoint written by Marcelo Aguilar and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-02-02 with Mathematics categories.
The authors present introductory material in algebraic topology from a novel point of view in using a homotopy-theoretic approach. This carefully written book can be read by any student who knows some topology, providing a useful method to quickly learn this novel homotopy-theoretic point of view of algebraic topology.
The Topology Of Fibre Bundles
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Author : Norman Earl Steenrod
language : en
Publisher:
Release Date : 1951
The Topology Of Fibre Bundles written by Norman Earl Steenrod and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1951 with Fiber bundles (Mathematics) categories.
Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory. This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject systematically. It begins with a general introduction to bundles, including such topics as differentiable manifolds and covering spaces. The author then provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles. The result is a classic and timeless work of great utility that will appeal to serious mathematicians and theoretical physicists alike.
Geometry Of Characteristic Classes
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Author : Shigeyuki Morita
language : en
Publisher: American Mathematical Soc.
Release Date : 2001
Geometry Of Characteristic Classes written by Shigeyuki Morita and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.
Characteristic classes are central to the modern study of the topology and geometry of manifolds. They were first introduced in topology, where, for instance, they could be used to define obstructions to the existence of certain fiber bundles. Characteristic classes were later defined (via the Chern-Weil theory) using connections on vector bundles, thus revealing their geometric side. In the late 1960s new theories arose that described still finer structures. Examples of the so-called secondary characteristic classes came from Chern-Simons invariants, Gelfand-Fuks cohomology, and the characteristic classes of flat bundles. The new techniques are particularly useful for the study of fiber bundles whose structure groups are not finite dimensional. The theory of characteristic classes of surface bundles is perhaps the most developed. Here the special geometry of surfaces allows one to connect this theory to the theory of moduli space of Riemann surfaces, i.e., Teichmüller theory. In this book Morita presents an introduction to the modern theories of characteristic classes.
Fibre Bundles
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Author : Dale Husemöller
language : en
Publisher: Springer Science & Business Media
Release Date : 1994
Fibre Bundles written by Dale Husemöller and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with Mathematics categories.
Basic properties, homotopy classification, and characteristic classes of fibre bundles have become an essential part of graduate mathematical education for students in geometry and mathematical physics. The new edition of this text includes two additional chapters, one on the gauge group of a bundle and the other on the differential forms representing characteristic classes of complex vector bundles on manifolds.