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Fiber Bundles And Homotopy
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Book Synopsis Fiber Bundles And Homotopy by : Dai Tamaki
Download or read book Fiber Bundles And Homotopy written by Dai Tamaki and published by World Scientific. This book was released on 2021-05-17 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Fibre Bundles written by D. Husemöller and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Topology of Fibre Bundles by : Norman Steenrod
Download or read book The Topology of Fibre Bundles written by Norman Steenrod and published by Princeton University Press. This book was released on 1999-04-25 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Topology of Fibre Bundles by : Norman Earl Steenrod
Download or read book The Topology of Fibre Bundles written by Norman Earl Steenrod and published by . This book was released on 1965 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Topology of Fibre Bundles by : Norman Earl Steenrod
Download or read book The Topology of Fibre Bundles written by Norman Earl Steenrod and published by . This book was released on 1951 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Modern Differential Geometry for Physicists by : Chris J. Isham
Download or read book Modern Differential Geometry for Physicists written by Chris J. Isham and published by Allied Publishers. This book was released on 2002 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author :Michael Charles Crabb Publisher :Springer Science & Business Media ISBN 13 :1447112652 Total Pages :344 pages Book Rating :4.4/5 (471 download)
Book Synopsis Fibrewise Homotopy Theory by : Michael Charles Crabb
Download or read book Fibrewise Homotopy Theory written by Michael Charles Crabb and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: Topology occupies a central position in modern mathematics, and the concept of the fibre bundle provides an appropriate framework for studying differential geometry. Fibrewise homotopy theory is a very large subject that has attracted a good deal of research in recent years. This book provides an overview of the subject as it stands at present.
Book Synopsis Lecture Notes in Algebraic Topology by : James F. Davis
Download or read book Lecture Notes in Algebraic Topology written by James F. Davis and published by American Mathematical Society. This book was released on 2023-05-22 with total page 385 pages. Available in PDF, EPUB and Kindle. Book excerpt: The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating. Moreover, by their second year of graduate studies, students must make the transition from understanding simple proofs line-by-line to understanding the overall structure of proofs of difficult theorems. To help students make this transition, the material in this book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated. Prerequisites for using this book include basic set-theoretic topology, the definition of CW-complexes, some knowledge of the fundamental group/covering space theory, and the construction of singular homology. Most of this material is briefly reviewed at the beginning of the book. The topics discussed by the authors include typical material for first- and second-year graduate courses. The core of the exposition consists of chapters on homotopy groups and on spectral sequences. There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic $K$-theory and the s-cobordism theorem. A unique feature of the book is the inclusion, at the end of each chapter, of several projects that require students to present proofs of substantial theorems and to write notes accompanying their explanations. Working on these projects allows students to grapple with the “big picture”, teaches them how to give mathematical lectures, and prepares them for participating in research seminars. The book is designed as a textbook for graduate students studying algebraic and geometric topology and homotopy theory. It will also be useful for students from other fields such as differential geometry, algebraic geometry, and homological algebra. The exposition in the text is clear; special cases are presented over complex general statements.
Download or read book Fibre Bundles written by Dale Husemöller and published by Springer Science & Business Media. This book was released on 1994 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis A Concise Course in Algebraic Topology by : J. P. May
Download or read book A Concise Course in Algebraic Topology written by J. P. May and published by University of Chicago Press. This book was released on 1999-09 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.
Book Synopsis Fibrations and Their Classification by : Petar Pavešić
Download or read book Fibrations and Their Classification written by Petar Pavešić and published by . This book was released on 2013 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Lectures On Algebraic Topology by : Haynes R Miller
Download or read book Lectures On Algebraic Topology written by Haynes R Miller and published by World Scientific. This book was released on 2021-09-20 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algebraic Topology and basic homotopy theory form a fundamental building block for much of modern mathematics. These lecture notes represent a culmination of many years of leading a two-semester course in this subject at MIT. The style is engaging and student-friendly, but precise. Every lecture is accompanied by exercises. It begins slowly in order to gather up students with a variety of backgrounds, but gains pace as the course progresses, and by the end the student has a command of all the basic techniques of classical homotopy theory.
Book Synopsis Cubical Homotopy Theory by : Brian A. Munson
Download or read book Cubical Homotopy Theory written by Brian A. Munson and published by Cambridge University Press. This book was released on 2015-10-06 with total page 649 pages. Available in PDF, EPUB and Kindle. Book excerpt: A modern, example-driven introduction to cubical diagrams and related topics such as homotopy limits and cosimplicial spaces.
Book Synopsis Introduction to Homotopy Theory by : Paul Selick
Download or read book Introduction to Homotopy Theory written by Paul Selick and published by American Mathematical Soc.. This book was released on 2008 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: Offers a summary for students and non-specialists who are interested in learning the basics of algebraic topology. This book covers fibrations and cofibrations, Hurewicz and cellular approximation theorems, topics in classical homotopy theory, simplicial sets, fiber bundles, Hopf algebras, and generalized homology and cohomology operations.
Book Synopsis Classifying Spaces and Fibrations by : J. Peter May
Download or read book Classifying Spaces and Fibrations written by J. Peter May and published by American Mathematical Soc.. This book was released on 1975 with total page 116 pages. Available in PDF, EPUB and Kindle. Book excerpt: The basic theory of fibrations is generalized to a context in which fibres, and maps on fibres, are constrained to lie in any preassigned category of spaces [script capital] F. Then axioms are placed on [script capital] F to allow the development of a theory of associated principal fibrations and, under several choices of additional hypotheses on [script capital] F, a classification theorem is proven for such fibrations.
Book Synopsis Basic Topology 3 by : Mahima Ranjan Adhikari
Download or read book Basic Topology 3 written by Mahima Ranjan Adhikari and published by Springer Nature. This book was released on 2023-03-15 with total page 488 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Mathematics For Physics: An Illustrated Handbook by : Adam Marsh
Download or read book Mathematics For Physics: An Illustrated Handbook written by Adam Marsh and published by World Scientific. This book was released on 2017-11-27 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique book complements traditional textbooks by providing a visual yet rigorous survey of the mathematics used in theoretical physics beyond that typically covered in undergraduate math and physics courses. The exposition is pedagogical but compact, and the emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints. Certain topics which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations, are avoided. The primary physical models targeted are general relativity, spinors, and gauge theories, with notable chapters on Riemannian geometry, Clifford algebras, and fiber bundles.
